Peter Hamburger, Gyorgy Petruska and Attila Sali- Saturated Chain Partitions in Ranked Partially Ordered Sets, and Non-Monotone Symmetric 11-Venn Diagrams

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Saturated Chain Partitions in Ranked Partially Ordered

Sets, and Non-Monotone Symmetric 11-Venn Diagrams

Peter Hamburger‡

Gyorgy Petruska†♭

and Attila Sali⋆

‡Department of Mathematical Sciences

†Department of Computer Science

Indiana University - Purdue University Fort Wayne

Fort Wayne, Indiana 46805

⋆

Alfred Renyi Institute of Mathematics⋆Budapest, P.O.Box 127 H-1364 Hungary

e-mail: ‡[email protected]

⋆[email protected]

♭[email protected]

Abstract

In this paper we show that there are at least 2110 non-isomorphic 11-doilies, that is,there are many non-isomorphic symmetric, non-simple, non-monotone 11-Venn dia-grams, with “many” vertices. We do not achieve the maximum vertex set size, 2046,but we approach it closely, improving from the previous 462 in [10] to 1837. The doiliesconstructed here cannot be constructed by either of the methods of [10] or [6]. Themain purpose of this paper is not to publish these attractive diagrams but to inspirenew studies by raising ideas, methods, questions, and conjectures, hoping for resultsanalogous to those generated in [10]. These ideas connect two seemingly distant areasof mathematics: a special area of combinatorial geometry, namely, certain families of simple closed Jordan curves in the plane, and the study of ranked partially ordered setsor posets.

1 IntroductionIn [10] the first (nonsimple) symmetric Venn diagram with 11 curves, that is an 11-doily, wasrecently published. This doily is special in many ways. It is monotone, (for the definitionsand the properties of ranked posets, as well as of the different type of doilies and Venn dia-grams, see Sections 1.1 and 1.2,) and thus it is isomorphic to a 11-doily which can be drawnwith all convex curves, see [1]. Its vertex set has the size 462 =

11

11−1

2

= κ11−1, the number

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of vertices in the middle layer of the 11-hypercube. This example is constructed by using asymmetric chain partition of the 11-hypercube. In [10] and [11] also a method is suggestedto settle this problem with these special properties, for any prime p greater than 11. Thiswas recently followed through in [6], by Griggs, Killian, and Savage, showing that indeed,there are nonsimple, monotone, symmetric p-Venn diagrams for every prime p greater than11 with the vertex set of size p

p−1

2 = κ p−1. This size of the vertex set is determined by the

fact that any symmetric chain partition of a ranked poset must have κ p−1 chains, (see [2]).Their proof is a constructive proof.

For p = 11 the maximal vertex set size (the simple doily case) is 211 −2 = 2046. The minimalpossible vertex set size, as allowed by Euler’s planar graph theorem and the rotational sym-metry, is 209. Though the cases with the vertex set size κ p−1 are settled, other cases, andin particular the most interesting case, the maximal-sized vertex set, that is the existence of simple doilies, is still unsolved for any prime number bigger than 7. The existence of simple

p-doilies with prime numbers greater than 7 is a open conjecture of Grunbaum [8]. Thereare also only a few known diagrams, with fewer than 462 vertices for p = 11, [12, 13].

Following the method suggested in [10, 11], Griggs at. el. in [6], with a delicate use of Greeneand Kleitman’s so-called parenthesis matching approach, (see [5]), showed that if p is anyprime number, then there is always a way to select a complete set of cyclic representatives of binary codes so that the induced subposet of the Boolean lattice (or the p-hypercube) has asymmetric chain partition such that it forms a “special” planar subgraph, (called a doodle) of the p-hypercube. The cyclic rotation of this graph generates a spanning, 2-connected planarsubgraph of the p-hypercube, (called a Venn model ). In [11] it is shown that this kind of graph is the dual graph of a non-simple symmetric Venn diagram.

In this paper we take another approach. Instead of using symmetric chain partitions of the

Boolean lattice we use saturated chain partitions. This leads us to have more faces in theVenn model, and thus many more vertices in the Venn diagram. The maximum numberof vertices that we could reach with this method so far is 1837 , which is 209 less than thepossible maximum 2046. But even in the minimal diagram we create in this paper there are627 faces in the Venn model and thus 627 vertices in the Venn diagram. This means that inaddition to the 462 symmetric chains, there are 198 additional saturated chains in the chainpartition. Many of them are in the upper half, and many are in the lower half of the lattice.This also means that these diagrams cannot be constructed with the method of [6], [10, 11],or [12, 13]. The diagrams are created in this paper are all non-monotone with many vertices.There are a few known non-monotone, symmetric 11-Venn diagrams. They are created by

another method using special path decomposition of the Boolean lattice, and they have onlyfew vertices, [12, 13].

We believe that these diagrams are interesting and definitely very attractive. Still the mainreason to publish this paper is not just showing those diagrams but rather presenting ideas,methods, conjectures, and problems. They can inspire further studies of ranked posets whichmay have important ramifications not only in the study of Venn diagrams but in other areas

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Ranked Posets

Boolean Lattice B n n-Hypercube Venn Diagram

n-Hypercube graph

Dual D(F)Venn GraphVenn Model

V(F)

Geometrical Graph

n-Doodle n-Doily

Figure 1: The logical relationship of Venn-related structures

as well. We also hope these methods can result in proving that there are p-doilies for allprime numbers p with all possible sizes of vertex sets bigger than κ p−1, and in particular, wehope they will lead us to a proof of Grunbaum’s conjecture on simple doilies.

The remainder of this paper is organized as follows. Section 1.1 describes the concepts of ranked posets, Boolean lattices, p-hypercubes, and some of their properties we use here. Theconcepts of chains, saturated chains, and symmetric chains are also defined. It also describesthe concept of symmetric or saturated chain partition of the Boolean lattice. Here also theconcepts of chain cover property, chain cover map, chain cover graph, chain cover tree, cyclicbinary code classes, poset of the cyclic binary code classes, the generator of the poset of the

cyclic binary code classes, and the (symmetric or saturated) chain cover generator graph arerecapitulated or introduced. Among them there are the new concepts such as the monotoneand non-monotone chain cover property, (but we simply call the second one the chain coverproperty,) extended chain cover graph, and the concept of conflict graph. Using these conceptsin Theorem 9, a necessary and sufficient condition for a saturated chain decomposition of the Boolean lattice with a planar embedding is given. This theorem generalizes the result of the main idea of [6], that is, Theorem 9 is the generalization of the result of Lemma 1 of [6].Section 1.2 describes the concept of Venn diagrams, it defines the different types of Venndiagrams. Here we also show the connections between the different type of Venn diagramsand the concepts listed in Section 1.1. The concept of symmetric Venn diagrams, that is,the concept of doilies is introduced in Section 1.3. This section also briefly repeats some of

the ideas and methods of [10] and [11] that are needed to understand the construction of doilies, and contains the key technical results about doilies. Also the connections betweenthe concepts listed in Section 1.1 and the concept of doily are studied. Sections 2 and 3contain the details of our construction of the many non-isomorphic, non-monotone 11-doilieswith different sizes of vertex sets, varying between 627 and 1837. Finally, Section 4 suggestssome ideas, methods, conjectures, and open problems.

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1.1 Ranked Posets and Chains

Definition 1. Let P = (P, ≤) be a poset. y ∈ P is said to cover x ∈ P if x < y and there isno z such that x < z < y. The poset P is ranked if there is an integer valued function r(x)on P such that r(x) = 0 for all minimal elements x of P and r(y) = r(x) + 1 for all x, y if y covers x. If P is ranked, then r(x) is called the rank of x, and the rank of P is maxxr(x).

For more information about ranked posets see [19].

One of the most important ranked posets is the Boolean lattice Bn. For a number n ≥ 0let [n] denote the set {1, 2, . . . , n}. The Boolean lattice Bn = (2[n], ⊆) is the ranked posetconsisting all subsets of [n] , ordered by inclusion. For s ∈ 2[n], r(s) is the cardinality of s. Itis convenient to view the elements of Bn as elements of {0, 1}n, the set of all n-bit strings.The order relation is defined by x ≤ y iff w(x) ⊆ w(y), where for x = x1x2 . . . xn ∈ {0, 1}n,w(x) = {i| xi = 1, 1 ≤ i ≤ n}. It is easily seen that |w(x)|, also called weight , is a rank on Bn.

The Boolean lattice Bn can also be viewed as a bipartite graph. This is the so-called n-cube

( n-hypercube). It has 2

n

vertices, they are the n-tuples of 0’s and 1’s, called the coordinatesof a vertex. Two vertices x and y with r(x) < r(y) are adjacent by an edge in the hypercubeif and only if they differ exactly by one coordinate, that is, y covers x. It is not hard to seethat the n-cube is a simple, connected, bipartite graph, in which each vertex has degree n.It is also known that in the n-cube there are n2n−1 edges, it is n-vertex- or edge -connected,and its diameter is n.

Definition 2. Let P be a ranked poset of rank n. The set {x ∈ P | r(x) = m} (0 ≤ m ≤ n)is called the m − level of the poset P . A finite subset C = {c1, c2, . . . , ck} in P is called a chain if ci < ci+1 for all i = 0, . . . , k − 1. A chain C = {c1, c2, . . . , ck} is a saturated chain iff it intersects every level between levels r(c1) and r(ck), that is if ci+1 covers ci for

all i = 0, . . . , k − 1. A saturated chain C = {c1, c2, . . . , ck} in P (X ) is called symmetric if r(c1) + r(ck) = n.

Clearly in a (symmetric) chain C = {c1, c2, . . . , ck} each i ranked element is covered by ai + 1-ranked element and it covers a i − 1-ranked element for all 2 ≤ i ≤ k − 1. We call thisproperty the monotone property of chains.

In particular, every symmetric chain has an element at level ⌊n/2⌋.

Definition 3. Let A = (A, ≤) be a finite ranked poset. A set C of chains is called a chainpartition of A = (A, ≤) or ( CP) iff each element of A belongs to one and only one chain of C. If each chain of C is a symmetric chain, then it is called a symmetric chain partition or

( SCP). If each chain of C is a saturated chain, then the partition is called a saturated chainpartition or ( SaCP) of A.

A useful concept is introduced in [6], that led Griggs, Killian, and Savage to find a patternthat gives a nonsimple, monotone, symmetric p-Venn diagram for any prime number p.

Definition 4. [6] Let C be a SCP in a finite ranked poset A = (A, ≤) and for each C ∈ C,let starter(C ) be the first element of C and let terminator(C ) be the last element of C. Call

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the longest chains in C the root chains. Say that C has the monotone1 chain cover propertyiff whenever C ∈ C and C is not a root chain, then there exists a chain φ(C ) ∈ C such that:

(*) starter(C ) covers an element φs(C ) of φ(C ) and

(**) terminator(C ) is covered by an element φt(C ) of φ(C ).

Call such a mapping φ a monotone chain cover mapping.

We introduce here a weaker version of Definition 4. We call this the chain cover property.

Definition 5. Let C be either a SCP or a SaCP in a finite ranked poset A = (A, ≤) and for each C ∈ C, let starter(C ) be the first element of C and let terminator(C ) be the last element of C. Call the longest chains in C the root chains. Say that C has the chain coverproperty iff whenever C ∈ C and C is not a root chain, then there exists a chain φ(C ) ∈ Cand elements φs(C ) and φt(C ) of φ(C ) such that:

(*) starter(C ) covers or is covered by φs(C ) of φ(C ),

(**) terminator(C ) covers or is covered by φt(C ) of φ(C ), and

(***) the rank of φs(C ) is smaller than the rank of φt(C ).

Call such a mapping φ a chain cover mapping.

Definition 6. Let C be a SCP or a SaCP in a finite ranked poset A = (A, ≤), and supposethat C has a unique root chain. In this case φ can be described by a rooted tree, T (C, φ),called a chain cover tree, in which each node corresponds to a chain C ∈ C and the parent of node C is φ(C ). If there is more than one root chain, we obtain a rooted forest.

Definition 7. Let C be a SCP or SaCP with the chain cover property for a poset A = (A, ≤).Let φ be a chain cover map for C. The chain cover graph, G(C, φ), is the graph whose verticesare the elements of A, and whose edges consist of the covering edges in the chains in Ctogether with the cover edges, for each non-root chain C ∈ C , from:

(*) starter(C ) to φs(C ) and

(**) terminator(C ) to φt(C ).

The extended chain cover graph eG(C, φ) is obtained from G(C, φ) by connecting the first

elements of the root chains to a special vertex b and the last elements of the root chains to a special vertex t, finally, adding the edge { b, t}.

1In[6] the word monotone is not used.

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Definition 8. Let us assume that a ranked poset A = (A, ≤) is given with an SaCP Cthat has the chain-cover property. For every C ∈ C the conflict graph coG(C ) is defined as follows. The vertex set of coG(C ) is the set of children of C , that is the set φ−1(C ).{C ′, C ′′} is an edge of coG(C ), if either φs(C ′) < φs(C ′′) < φt(C ′) < φt(C ′′) or φs(C ′′) <φs(C ′) < φt(C ′′) < φt(C ′).

The following theorem is a generalization of Lemma 1 of [6].

Theorem 9. Let A = (A, ≤) be a ranked poset with an SaCP C that has the chain-cover property. The extended chain cover graph eG(C, φ) has a planar embedding iff the conflict graph coG(C ) is bipartite for all C ∈ C .

Proof. Let us assume first, that eG(C, φ) has a planar embedding. For every non-root chainC ∈ C the part of φ(C ) between φs(C ) and φt(C ) together with C and the covering edgesbetween starter(C ) to φs(C ) and terminator(C ) to φt(C ) form a cycle in the graph. Eachchild of C must go entirely inside or entirely outside of this cycle in the planar embedding.For a root chain C r the edges between C r and b and t, respectively, and the edge { b, t} form

a cycle that have the same property as above, that is child of C r must go entirely insideor outside of it. It is easy to see, that if C ′ and C ′′ are children of C , and they both goinside (resp. outside) of the cycle, then there is no edge between them in the conflict graphcoG(C ). That is, the vertices of coG(C ) are properly colored with colors {inside, outside}.

In order to prove the converse, let us assume that all graphs coG(C ) are bipartite. Thechains in C form the rooted forest T (C, φ). Without loss of generality we may assume thatit is a rooted tree (otherwise we could work componentwise). Use an arbitrary post-orderwalk of T (C, φ), that is an ordering of the chains in C with the property that children of C come before C , for all C ∈ C. The planar embedding is constructed by induction followingthis order. Consider C ∈ C. Since coG(C ) is bipartite, the children of C can be parti-

tioned into two classes, such that for any two chains C ′, C ′′ in the same class the intervals(φs(C ′), φt(C ′)) and (φs(C ′′), φt(C ′′)) in chain C either have no common inner point, or oneof them contains the other. (Like non-crossing families.) That is, if we think φs(C ′)’s asleft parentheses, and φt(C ′)’s as right parentheses, then we obtain a proper parenthesization,where φs(C ′) is matched with φt(C ′), for all C ′. Using that, starting from the innermostpairs of matched parentheses we can embed the children of C in one class to the left sideof C , and the children in the other (color) class on the right side of C together with theirdescendants, as shown on Figure 2. Here we assume that the planar embedding of a childchain and all its descendants is drawn inside a crescent.

Note, that in [6] the special case of symmetric chain decomposition with a monotone chaincover property is used. In that case, all graphs coG(C ) are edgeless, that is trivially bipartite.We used the concept of extended chain cover graph eG(C, φ), since in the applications the

vertices (0, 0, . . . , 0) and (1, 1, . . . , 1) play the role of b and t in the constructions of p-doodles.

The following theorem is shown in [2].

Theorem 10. The Boolean lattice Bn has a SCP.

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Figure 2: Embedding the descendants of a chain

In [6] another proof of this theorem is given. This SCP is called the Greene-Kleitman SCP,

see [5]. In any SCP every symmetric chain meets the level X

⌊n/2⌋

in one set, thus there mustbe

n⌊n/2⌋

non-empty symmetric chains in the partition.

It is also shown in [6],

Lemma 11. The Greene-Kleitman SC P of Bn has the monotone chain cover property for all positive integer numbers n.

Definition 12. If x = x1, x2, . . . , xn−1, xn ∈ {0, 1}n we define a shift of x by s(x) =xn, x1, x2, . . . , xn−1. A rotation of x is a composition of shifts, that is, let s1(x) = s(x) and

for i > 1, let si(x) = s(si−1(x)). Denote the set of all rotations of all elements of the Boolean lattice Bn by S n(Bn) = S n.

Definition 13. Define the relation “ ∼” on {0, 1}n by x ∼ y if y = si(x) for some integer i ≤ 0. Then ∼ is an equivalence relation on {0, 1}n and the equivalences classes are called cyclic binary code classes, or CBCC. An element x of a class X of CBCC is called a repre-sentative of the class X .

Definition 14. Let N n be the CBCC of {0, 1}n and define the poset of the cyclic binarycode classes, (or PCBCC), N n, by N n = ( N n, ) with a transitive ordering generated by the rule a1 a2 if some x ∈ a1 differs from some y ∈ a2 only in one bit i, where xi = 0 and yi = 1.

Definition 15. [11] Let B be a subset the Boolean lattice Bn. An element x (not necessarily in B) of Bn is called independent from B, if x cannot be obtained by a shift or a rotation of any other element of B. A subset B of elements of Bn is called independent if every element x of B is independent from B.

Definition 16. [11] A generator G is a maximal independent subset of the Boolean latticeBn with 1 ≤ w(x) ≤ n − 1, for every x ∈ G .

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Remark 17. It is well known that if p is a prime number, then a set of representatives of elements of the PCBCC G = N p − {0, 0, . . . , 0, 1, 1, . . . , 1} is a generator of the Boolean lattice {0, 1}n, and in a generator G there are exactly

λk =

pk

p

elements with weight k, for each 1 ≤ k ≤ p − 1, and thus, there are

λ =

p−1k=1

λk

elements.

Definition 18. Let p be a prime number and let a set of representatives of elements of G = N p − {0, 0, . . . , 0, 1, 1, . . . , 1} be a generator for the Boolean lattice B p. Let C be a CP of G with the chain cover property and with the chain cover map φ. The chain cover graph G(G, φ) is called a chain cover generator graph (or CCGG). If each chain in C is

a symmetric chain, then the graph is called a symmetric chain cover generator graph, (or SCCGG.) If each chain in C is a saturated chain, then the graph is called a saturated chaincover generator graph, (or SaCCGG).

The main result in [6] is the following

Theorem 19. For every prime number p there is a SCCGG.

1.2 Venn diagrams

A Venn diagram or an n-Venn diagram consists of n simple closed Jordan curves F =

{C 1, C 2, . . . , C n} in the plane such that X 1

X 2

· · ·

X n is a nonempty, connected, openset (region) (called a face); here, X i is either the bounded interior or the unbounded exteriorof C i, i = 1, 2, . . . , n . If f is a face and ∂f

C i = ∅ is a simple arc, then it is called an edge

of the face f. It is known that the boundary of a face consists of finitely many edges. A facewith k edges (0 < k ≤ n) will be called a k-face. One can put various restrictions on thediagrams and obtain special classes of diagrams. We follow Grunbaum [7] in the terminology.

We note that each of the 2n faces can be described by an n-tuple of zeros and ones where theith coordinate is a 0 if X i is the unbounded exterior of C i, otherwise it is 1, i = 1, 2, . . . , n .It is clear that there is a one-to-one correspondence between the 2n faces of a Venn diagramand the vertices of the n-dimensional hypercube. If A = X 1X 2 · · ·X n is a face of aVenn diagram then the corresponding n-tuple in the hypercube is called the description of A. The weight of a face is the weight of the corresponding description.

A Venn diagram is simple if at most two of its curves intersect (transversely) at any pointin the plane. Among the nonsimple Venn diagrams, we will consider only those in whichany two curves meet (not necessarily transversely) in isolated points. With this restrictionon the Venn diagrams it is easy to see that if two faces A and B in a Venn diagram share a

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common edge, then their descriptions must differ exactly in one coordinate. In this case thefaces are called adjacent. Also it is not hard to see that if two faces A and B differ in theirdescriptions in two or more coordinates then the faces cannot share a segment of a curve astheir common boundary.

Two Venn diagrams are isomorphic if, by a continuous deformation of the plane, one of them

can be changed into the other or its mirror image.

A Venn diagram is monotone if for every k, a face with weight k is adjacent to at least oneface with weight (k − 1) (for 0 < k) and to at least one face with weight (k + 1) (for k < n).In [1] it was shown that a Venn diagram is isomorphic to a convex Venn diagram (a diagramof all convex curves), if and only if the diagram is monotone. A. Renyi, K. Renyi, and J.Suranyi in [16] showed that for every positive integer n there is a convex Venn diagram, andthus there is a monotone one as well.

To each Venn diagram one can associate two graphs, see [3]. The Venn diagram itself canbe viewed as a planar graph V (F ) where all the intersection points of the curves in F arethe vertices of V (F ) and the edges of the faces are the edges of V (F ). In proper context,confusion rarely arises from also calling this graph a Venn diagram. In the rest of this paperthe notations F and V (F ) are freely interchanged. The Venn diagram V (F ) may have mul-tiple edges. The planar graph dual to V (F ) will be called the Venn graph, denoted by D(F ).

Quite often we will consider the Venn graph D(F ) to be superimposed on the Venn diagramin the plane, and we will use descriptive statements such as “the curve C of F intersects orcrosses the edge e of D(F )”.

Several interesting properties of Venn diagrams and Venn graphs were derived in [3]. Here

we simply state those properties we need as remarks and refer the reader to [3] for proofs.

Remark 1 The Venn graph D(F ) of a Venn diagram V (F ) is a planar, spanningsubgraph of the |F|-hypercube.

Remark 2 No two edges of a face of a Venn diagram belong to the same curve.

Remark 3 A Venn graph D(F ) is simple, and 2-connected, but the deletion of any pair of adjacent vertices does not disconnect the graph.

Remark 4 If F is a simple Venn diagram, then each face of D(F ) is a quadrilat-eral, and hence D(F ) is a maximal bipartite planar graph.

Remark 5 The Venn graph D(F ) of a simple Venn diagram has connectivity 3.

In [11], the Venn graph has been extensively used to study and construct Venn diagrams.Given a planar graph that meets the conditions described below, an associated Venn diagramcan be constructed, such that the corresponding Venn graph is isomorphic to the originalgraph.

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Definition 20. [11] Let G∗ be a 2-connected, planar, labeled, spanning subgraph of the n-hypercube. G∗ is called a Venn model if

1. To each edge e of G∗ we assign the index (called the edge number) of the coordinatewhere the descriptions of the two end-vertices of e differ;

2. Any two faces of G∗ share at most one edge with a given edge number; and

3. An edge number that appears on the boundary of a face of G∗ must appear exactly twice on that face.

G∗ is called a simple Venn model if each face of G∗ is a 4-face and the graph G∗ is 3-connected.

It is easily seen that every Venn graph D(F ) is a Venn model. The construction below showsthat the converse statement holds true as well.

Construction 21. [11] If G∗ is the Venn model and N is an edge numbering with the above

properties, then a Venn diagram F can be constructed in the following way.Step 1. Create a graph D∗(G∗) as follows. Place a vertex xF in each face F of G∗,and join it to each edge of F by a simple Jordan arc, such that

(i) The arcs inside in a face F meet only at the new vertex xF , and

(ii) In every edge of G∗ exactly two arcs meet.

Step 2. Assign to each simple Jordan arc the edge number of the edge it meets. Thisidentifies each simple closed Jordan curve in the Venn diagram.

Step 3. The set of simple closed Jordan curves created by the procedure above is F .

It is easy to see that the diagram F thus obtained is a Venn diagram, and if for some Venndiagram F , G∗ = D(F ), then D∗(D(F )), is graph-isomorphic to the Venn diagram F .

Now we show

Lemma 22. Let C be a SCP or a SaCP with the monotone chain cover property for a poset A = (A, ≤), and let φ be a chain cover map for C. Then a planar embedding P (C, φ) of thechain cover graph G(C, φ) is a Venn model.

Proof. Theorem 9 says that the chain cover graph G(C, φ) has a planar embedding P (C, φ).First we show that P (C, φ) satisfies the conditions of Definition 20. Indeed, if an elementc1 of a chain C ∈ C covers an element c2 either in the chain C, or c2 is the terminator(C )of the chain C and it is covered by the element c1 = φt(C ), then they differ exactly by onecoordinate, say i, 1 ≤ i ≤ n. Similarly, if c1 is the starter(C ) of C and it covers an elementc2 = φs(C ), then they differ exactly by one coordinate, say i, 1 ≤ i ≤ n. Designate the edgenumber i to the edge between c1 and c2. This gives the edge numbering N of the edges of P (C, φ). The set N satisfies the conditions of Definition 20. Indeed, since between two facesthe boundary is a chain therefore each edge number appears exactly once. The boundary of

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a face contains two chains intersecting twice at two vertices in two different levels, thus anedge number that appears in a face of P (C, φ) appears exactly twice in that face, once ineach chain. Since each vertex of P (C, φ) lies on a cycle, the graph P (C, φ) is 2-connected.Finally, according to Construction 21, a diagram F can be constructed by using P (C, φ) asthe Venn model.

If C is a SCP or a SaCP with the chain cover property for a poset A = (A, ≤), and φ is achain cover map for C, then a planar embedding P (C, φ) of the chain cover graph G(C, φ)is not necessarily a Venn model. Indeed, property 2. of Definition 20 can be violated if thecover edges, the two edges connecting a chain to its parent have the same edge number, seechain C 10 of Table 2 and Figure 6. Let f 1 and f 2 be two faces, and suppose that e1, e2 arecover edges of a chain and they are on the common boundary of these faces with the sameedge number, say i. These edges are called parallel cover edges. Note that property 2. of Definition 20 can be violated by the following two other types of chains. If the cover edgesof a chain have different edge numbers but these cover edges are in reversed order, that is,starter(C ) is covered and terminator(C ) covers an element of the parent chain, respectively,then the cover edges are called reversed cover edges, see chain C

47of Table 4 and Figure 6.

If the cover edges have different edge numbers but one of them, either starter(C ) is covered,or terminator(C ) covers an element of the parent chain, respectively, then the cover edgesare called half reversed cover edges, see chain C 45 of Table 4 and Figure 6. Note also thatonly these kind of chains with cover edges above can violate property 2. of Definition 20.Sometimes this can be corrected with the following procedure.

Definition 23. Let P (C, φ) be a planar embedding of a chain cover graph G(C, φ) that violatesproperty 2. of Definition 20. Suppose that either e1, e2 are parallel cover edges of a chain with the same edge number, say i, or they are reversed cover edges, or they are half reversed cover edges, and suppose that they are on the common boundary of f 1 and f 2 faces. Let x1

and x2 be two nonadjacent vertices of P (C, φ) and suppose that their binary representation differ exactly in one bit, say j, and j = i. The edge {x1, x2} is said to be a crossing edge if the addition of this edge to the planar graph P (C, φ) separates one of the faces into two faces,and in the newly created planar graph the edges e1 and e2 are not on the common boundary of any two faces.

Construction 24.

Suppose that the planar embedding P (C, φ) of a chain cover graph G(C, φ) violates property2. of Definition 20. Choose a set of crossing edges, and add them one by one preserving theplanarity of graph P (C, φ) until each parallel cover edge pair is separated, and each violationof property 2. of Definition 20 by reversed or half reversed cover edges are corrected. Thenewly constructed planar graph is a Venn model. Now some of the old edges of the graphP (C, φ) can be deleted without violating Definition 20. Delete them until you cannot deleteany more old edge without violating Definition 20. The obtained planar graph P ∗(C, φ) iscalled a minimal Venn model.

This procedure is illustrated in Lemma 38 and Figures 6 and 8.

11



Using Euler’s Theorem for planar graphs |F | + |V | − |E | = 2 (where |F | is the number of faces, |V | is the number of vertices, and |E | is the number of edges of the graph), an easyargument ([10]) shows that the number of vertices |V | of a Venn diagram with n curvessatisfies the inequality

⌈2n − 2

n − 1⌉ ≤ |V | ≤ 2n − 2.

1.3 Symmetric Venn diagrams

A Venn diagram of n curves is said to be symmetric if a rotation through 360/n degreesmap the family of curves onto itself, that is, the diagram is not changed by the rotation. Itis clear that curves of a symmetric Venn diagram can be obtained by n successive rotationsthrough 360/n degrees of any of the curves. It is not hard to see that an n-Venn diagram issymmetric iff the descriptions of two faces are rotations (cyclic permutations) of each other,then the faces are congruent. In particular, choosing a representative of each congruenceclass, a symmetric n-Venn diagram can be reconstructed by rotating successively the cho-sen faces about one given point through angles of 360/n degrees [15]. A symmetric Venn

(n-Venn) diagram will be called a doily (n-doily ).

Symmetric Venn diagrams have been studied by several researchers including Henderson [15],Grunbaum [8, 9], Schwenk [18], Edwards [4], Ruskey [17], Hamburger [10, 11, 14], Hamburgerand Sali [12, 13], and recently Griggs, Killian, and Savage [6]. For the brief history of resultson symmetric Venn diagrams see [10]. It is known (see [15]) that symmetric Venn diagramsdo not exist for composite numbers n.

Let p be an arbitrary fixed prime number. By the rotational symmetry, the cardinality of the vertex set |V | of a p-doily must meet the following conditions [11]

(i) ⌈

2p−2

p−1 ⌉ ≤ |V | ≤ 2

p

− 2,(ii) p is a divisor of |V |.

In the trivial case p = 2, |V | = 2 is uniquely determined, the conditions are obvious. In whatfollows we assume p ≥ 3. Properties (i) and (ii) obviously hold for κ p−1, and by Fermat’slittle theorem, 2 p−1 ≡ 1 (mod p), also 2 p − 2 satisfies both conditions. The first numberprovides non-simple p-doily for every prime number p, and the second number is a candidatefor |V | of a simple p-doily, but only known to exist for the cases p = 3, 5, 7.

Notation 25. We denote by

K p =

p p−12

,

p p−12

+ p,

p p−12

+ 2 p , . . . , 2 p − 2

= {κ p−1, κ p−1+ p,κ p−1+2 p , . . . , κ2p−2},

the set of numbers between the two numbers mentioned above and satisfying (i) and (ii).Also, let

Λ p =

pp−1

2

p

,

pp−1

2

p

+ 1,

pp−1

2

p

+ 2, . . . ,2 p − 2

p

= {λ p−1, λ p−1 + 1, λ p−1 + 2, . . . , λ2p−2}.

12



Again, for p = 3, 5, 7 every element of K represents |V | for a suitably constructed doily [11].Our main topic in this paper is to explore the case p = 11. In [10] the case κ11−1 = 462and in [11] the cases 462, 462 + 11, 462 + 22, . . . , 462 + 54 · 11 = 1001 are settled. Inthis paper applying another method based on saturated chains we construct 11-doilies for627, 627 + 11, 627 + 22, . . . , 627 + 110 · 11 = 1837. In our next paper we return to the prob-lem of the remaining numbers between 1837 and 211 − 2 = 2046. It is remarkable that this

gap 2046 − 1837 = 209 is exactly the cardinality of the conjectured minimal 11-doily. Thecases 220, 220 + 11, . . . , 220 + 24 · 11 = 484 are settled in [14] based on a third approach,however the single remaining case 209 eluded us so far. In [6] the cardinality κ p−1 for primenumbers p ≥ 11 have been settled. It can be seen that their construction extends to thecases 462+11, 462+22, . . . , 1276, however that construction does not seem to go beyond 1276.

Definition 26. [11] A p-doodle is a subgraph G of the p-hypercube with the following prop-erties:

1. The set G of descriptions of the vertex set G is a generator set,

2. {s(a) | a ∈ G, s ∈ S p} ∪ {0, 0, . . . , 0, 1, 1, . . . , 1} is a set of descriptions of a 2-connected, spanning, planar subgraph of the p-hypercube, where S p is defined in Definition 12.

Note that the elements of set Λ p are possible number of faces of a p-doodle.

The following procedure describes one of the methods of [11]. This shows how can a p-Venn model of a p-doily be constructed from a p-doodle. This is illustrated by Figures 8 and 9, aswell as Figures 13 and 14.

Construction 27.Let G be a doodle, and let C 1 be a 2π/p sector of a circle C with center c. Let H be a planarembedding of the graph G into the interior of C 1. We add as vertices the center point c andthe infinite point c∗ to the graph H. We may suppose that the vertex having weight p − 1of the graph H can be connected to c and the vertex with weight 1 can be connected toc∗ preserving the planar graph property. Adding these edges we denote the extended graphby G0. We assign the p-tuples 1, 1, . . . , 1 and 0, 0, . . . , 0 to the vertices c and c∗ as theirdescriptions, respectively. These descriptions are also denoted by c and c∗. We associate toshift s = s1 the rotation of the plane by angle α = 2π/p around c, and the rotation is alsodenoted by s1. The rotated copies of G0 are denoted by

G j = s j(G0), (0 ≤ j < p).

Given a vertex s j(v) ∈ G j , v ∈ G0, we assign the p-tuple s j(d) to s j(v) as its description if and only if d is the description of the vertex v. Note that s j(c) = c, s j(c∗) = c∗. Let the graphD(G) = ∪ p−1

j=0G j, where e is an edge of D(G) if and only if it is an edge of some G j . This graphD(G) is graph-isomorphic to a 2-connected, spanning, labeled subgraph of the p-hypercube.Assign a number between 1 and p to each edge of D(G) corresponding to the coordinate

13



where the descriptions of the two end-vertices of the edge differ. If the graph D(G) satisfiesall conditions of Definition 20, then the created graph is called the p-Venn model of a p-doily .

Using the method of Construction 21 with the Venn model of a doily, a Venn diagram isconstructed. This diagram is graph-isomorphic to a doily.

Lemma 28. Let p be a prime number and let a set of representatives of elements of G = N p − {0, 0, . . . , 0, 1, 1, . . . , 1} be a generator for the Boolean lattice B p. Let C be a CP of G with the monotone chain cover property and with the chain cover map φ, and supposethat each conflict graph is bipartite. Then the planar embedding P (C, φ) of the chain cover graph G(G, φ) with that transformation set S p(B p) (see Definition 12) is a p-doodle of a p-doily. Suppose that C is a CP of G with the chain cover property and it has a minimal Venn model V (C, φ), then the minimal Venn model V (C, φ) with that transformation set S p(B p) isa p-doodle of a p-doily.

Proof. Remark 17, Lemma 22 shows that the planar embedding P (C, φ) of the chain

cover graph G(G, φ) with the set S p(B p) is a p-doodle. Finally, if this p-doodle is used inConstruction 27, then a p-doily is obtained.

Remark 29. Let p be a prime number, and let a set of representatives of elements of G = N p − {0, 0, . . . , 0, 1, 1, . . . , 1} be a generator for the Boolean lattice B p. Let C be a CP of G with the monotone chain cover property and with the chain cover map φ. Since each chain C has the monotone property, thus any p-doily which is received from the p-doodle described in Lemma 28 is a monotone Venn diagram, and thus it is isomorphic to a p-doily which can be drawn with all convex curves.

Remark 30. A Venn diagram constructed from a minimal Venn model may or may not be

monotone. All Venn diagrams created below are nonmonotone.Recall that the vertices, edges, and faces of a Venn model correspond to the faces, edges, andvertices of the Venn diagram, respectively. Note that some of the pairs of vertices in a graphof a doodle may be connected by an edge, without violating the conditions of Definition 20,obtaining a new graph of a doodle. Every time a set of new edges is added to the graph of adoodle, the number of edges of the corresponding Venn model is increased by a multiple of

p. This increases number of vertices of the corresponding doily by a multiple of p. Note alsothat in the Venn model of a doily, sometimes p symmetric pairs of vertices from differentimages of G1 can be connected by p edges without violating the conditions of Definition 20.Adding these p edges, a new Venn model of a doily is created. This also increases the number

of vertices of the corresponding doily by p. This procedure is illustrated in Figures 13 and 14.

Definition 31. [11] A doodle or a Venn model is called extendable if some pairs of thevertices of the graph of the doodle can be connected by an edge, and/or p symmetric pairsof vertices from different images of G1 of the corresponding Venn model can be connected by p edges without violating the conditions of Definition 20. It is fully extendable if it isextendable, and the resulting Venn model of a doily is a 3-connected graph having all 4-faces.

14



Let p be a fixed prime number.

Definition 32. A p-Venn model G of a doodle of a p-doily G∗ is called minimal if after deletion of any edge e of G the rotation of G p-many times does not result in a p-Venn model G∗

1. A p-Venn model of a p-doily is called minimal if it is constructed as it is described in Construction 27 from a minimal p-Venn model G.

Definition 33. A minimal p-Venn model of a doodle G is called chain-minimal if it is theedge disjoint union of saturated chains. A chain-minimal p-Venn model is called minimumchain-minimal if its cardinality is κ p−1, that is, if each chain is a symmetric chain.

Every p-Venn model which is created from a SCP of the Boolean lattice is minimum chain-minimal, and thus according to [6] there is a minimum chain-minimal p-doodle for all primenumbers p.

Now, our approach is the following. Given a prime number p, and an integer m ∈ Λ p, finda chain-minimal SaCCGG G, |G(L)| = m (where G(L) is the number of faces of G), applythe shift p − 1 times getting a p-Venn model, and then extend it as much as possible, hopingthat it is fully extendable. In the next section we prove the following

Theorem 34. For p = 11 and 60 ∈ Λ11 there exists a chain-minimal SaCCGG G, |G(L)| =60. This creates a minimal 11-Venn model with 627 faces, and thus an 11-doily with 627vertices.

In Section 3 we show

Theorem 35. The minimal 11-Venn model of Theorem 34 is extendable with 110 edges toa Venn model of 1837 faces.

The proof of Theorem 34 is given in the next section in a series of lemmas and by an explicittable of vertices. The proof of Theorem 35 relies upon the extension of the underlying 11-doodle by adding 96 internal edges and 11 external edges. The details are shown in thefigures.

2 A chain-minimal saturated chain cover generator graph

(SaCCGG) with 60 saturated chains, for an 11-doilywith 627 vertices.

To create a p-doily we follow the following procedure: First we choose a generator, a set of binary p-tuples. Next we partition this set into the set of saturated chains. Now we specifya chain cover map φ. We check whether φ satisfies the chain cover property, or the monotonechain cover property. If one of them holds, then from the set of saturated chains with thehelp of the chain cover map φ we create a chain cover tree, and all of its conflict graphs.Using Theorem 9 and the chain cover tree we check whether the set of saturated chains hasa planar embedding. If the answer is affirmative, then we create an extended chain covergraph, and its planar drawing. Now we check whether property 2. of Definition 20 is satisfied.

15



If not we add crossing edges until it is corrected. (If property 2. is satisfied, then we skipthis step.) If it is possible, then we delete some old edges without violating Definition 20in order to created a minimal Venn model of a doodle. From the minimal Venn model of adoodle we create a minimal Venn model of a doily. Finally from the minimal Venn model of a doily the doily itself is created, and the required Jordan curve can be obtained.

After then we add edges to the minimal Venn model of a doodle, if it is possible, adding themone by one, or any subset of edges, to create other doodles, and thus other nonisomorphicdoilies. We do this until a maximal Venn model of a doodle created, thus a doily with amaximal vertex set. This procedure is illustrated for p = 11 in the next two sections.

Now we choose a set of binary 11-tuples, a generator, see Table 1. Note that finding agenerator for any fixed prime integer p is algorithmically determined; it is similar to par-tition problems that have been already been studied in combinatorics and number theory.Thus finding all doodles for any fixed prime integer p with an exhaustive search is alsodetermined. Hence finding a suitable doodle for any fixed prime number p is determinedas well. We do not know any algorithm to find one desired doodle without finding all of them.

It is tedious but not hard to check that the set of 11-tuples in Table 1 satisfies the conditionsof a generator. (In order to make Table 1 shorter we write the binary 11-tuples by indicatingthe locations of 1’s and we omit the 0’s. For example {1, 6} corresponds to 1000010000.By doing this it is easier to locate the binary 11-tuples in Figure 8.) The edge numbers of the Venn model of the doodle can be seen in Figure 6. Recall that an edge number is thelocation of the new 1 in the end vertex, as generated by the binary codes of Table 1. Fromthe generator we create the Venn model of the doodle.

Lemma 36. For p = 11 there exists a SaCP of the 11-hypercube with 60 saturated chains.There is a chain cover map φ, this chain cover map satisfies the chain cover property.

Proof. Tables 2,3 and 4 show the representatives of the cyclic binary code classes partitionedinto 60 saturated chains. In the tables the φ(C ), φs(C ), and φt(C ) denote the parent of thechain C , the first element of the parent chain and the second element of the parent chainthat connect the chain to the parent chain, respectively.

It is easy to check that the binary 11-tuples of Table 1 correspond to the SaCP of Tables 3, 2and 4.

Figure 3 shows the chain cover tree with 60 saturated chains.

Lemma 37. The conflict graphs of the chain cover tree of Figure 3 are all bipartite and thusthe extended chain cover graph is planar.

Proof. From the chain cover tree it is easy to see that there are only a few conflict trees,and they have only few edges. Figures 4 and 5 show the conflict graphs of the chain cover

16



w(v) = 1 # w(v) = 10 # w(v) = 3 # w(v) = 5 # w(v) = 6 #{1} 1 {1-10} 186 {1,6,3} 7 {1,6,3,7,9} 52 {1,6,3,7,9,5} 94w(v) = 2 # w(v) = 9 # {1,3,8} 8 {1,6,3,4,7} 53 {1,6,3,7,9,2} 95{1,6} 2 {1,6,3,4,7,5,2,8,9} 181 {1,3,9} 9 {1,6,3,10,4} 54 {1,6,3,7,9,4} 96{1,3} 3 {1,3,8,6,2,7,9,10,4} 182 {1,3,5} 10 {1,6,3,10,7} 55 {1,6,3,4,7,5} 97{1,5} 4 {1,3,9,4,6,5,8,2,10} 183 {1,5,6} 11 {1,3,8,6,10} 56 {1,6,3,10,7,4} 98

{1,9} 5 {1,9,5,10,2,6,7,4,8} 184 {1,5,2} 12 {1,3,8,6,7} 57 {1,3,8,6,10,7} 99{1,2} 6 {1,2,6,4,8,3,7,10,5} 185 {1,5,10} 13 {1,3,8,6,9} 58 {1,3,8,6,7,4} 10w(v) = 4 # w(v) = 7 # {1,9,5} 14 {1,3,8,6,2} 59 {1,3,8,6,7,9} 10{1,6,3,7} 22 {1,6,3,7,9,5,2} 136 {1,9,6} 15 {1,3,8,10,2} 60 {1,3,8,6,9,10} 10{1,6,3,4} 23 {1,6,3,7,9,2,4} 137 {1,9,10} 16 {1,3,8,10,9} 61 {1,3,8,6,2,9} 10{1,6,3,10} 24 {1,6,3,7,9,4,5} 138 {1,9,2} 17 {1,3,8,9,2} 62 {1,3,8,6,2,7} 10{1,3,8,6} 25 {1,6,3,4,7,5,2} 139 {1,2,10} 18 {1,3,9,2,10} 63 {1,3,8,10,2,6} 10{1,3,8,2} 26 {1,6,3,10,7,4,5} 140 {1,2,6} 19 {1,3,9,2,6} 64 {1,3,8,10,9,2} 10{1,3,8,10} 27 {1,3,8,6,10,7,4} 141 {1,2,3} 20 {1,3,9,4,2} 65 {1,3,9,2,6,10} 10{1,3,8,9} 28 {1,3,8,6,7,4,9} 142 {1,2,7} 21 {1,3,9,4,8} 66 {1,3,9,2,6,4} 10{1,3,9,2} 29 {1,3,8,6,7,9,10} 143 w(v) = 8 # {1,3,9,4,6} 67 {1,3,9,4,2,10} 10{1,3,9,4} 30 {1,3,8,6,9,10,2} 144 {1,6,3,7,9,2,4,5} 166 {1,3,9,6,5} 68 {1,3,9,4,8,2} 11{1,3,9,6} 31 {1,3,8,6,2,7,9} 145 {1,6,3,4,7,5,2,8} 167 {1,3,9,5,4} 69 {1,3,9,4,8,10} 11{1,3,9,5} 32 {1,3,9,2,6,10,4} 146 {1,6,3,10,7,4,5,2} 168 {1,3,9,5,2} 70 {1,3,9,4,8,5} 11{1,3,5,2} 33 {1,3,9,4,2,10,5} 147 {1,3,8,6,10,7,4,9} 169 {1,3,5,2,8} 71 {1,3,9,4,6,8} 11{1,3,5,4} 34 {1,3,9,4,8,2,10} 148 {1,3,8,6,2,7,9,10} 170 {1,3,5,2,4} 72 {1,3,9,4,6,5} 11{1,3,5,6} 35 {1,3,9,4,8,5,10} 149 {1,3,9,2,6,10,4,8} 171 {1,3,5,4,8} 73 {1,3,9,5,4,2} 11{1,5,6,10} 36 {1,3,9,4,6,5,8} 150 {1,3,9,4,8,2,10,5} 172 {1,3,5,4,6} 74 {1,3,9,5,2,8} 11{1,5,6,2} 37 {1,3,9,5,2,8,4} 151 {1,3,9,4,6,5,8,2} 173 {1,3,5,6,10} 75 {1,3,5,2,4,8} 11{1,5,2,9} 38 {1,3,5,4,6,8,10} 152 {1,3,5,4,6,8,10,2} 174 {1,5,6,10,2} 76 {1,3,5,4,6,8} 11{1,5,10,2} 39 {1,3,5,6,10,2,4} 153 {1,3,5,6,10,2,4,9} 175 {1,5,6,2,3} 77 {1,3,5,6,10,4} 11

{1,9,5,10} 40 {1,5,6,2,3,4,9} 154 {1,9,5,10,2,6,4,8} 176 {1,5,2,9,6} 78 {1,3,5,6,10,2} 12{1,9,6,5} 41 {1,5,6,2,3,9,10} 155 {1,9,5,10,2,6,7,4} 177 {1,9,5,10,2} 79 {1,5,6,2,3,4} 12{1,9,10,6} 42 {1,9,5,10,2,6,4} 156 {1,9,2,10,6,4,8,7} 178 {1,9,6,5,10} 80 {1,5,6,2,3,9} 12{1,9,2,10} 43 {1,9,5,10,2,6,7} 157 {1,2,6,4,8,9,7,3} 179 {1,9,10,6,7} 81 {1,9,5,10,2,6} 12{1,2,10,6} 44 {1,9,6,5,10,7,8} 158 {1,2,6,4,8,3,7,10} 180 {1,9,10,6,8} 82 {1,9,6,5,10,8} 12{1,2,6,9} 45 {1,9,10,6,8,7,2} 159 {1,9,2,10,6} 83 {1,9,6,5,10,7} 12{1,2,6,4} 46 {1,9,2,10,6,7,4} 160 {1,2,6,9,4} 84 {1,9,10,6,8,7} 12{1,2,6,3} 47 {1,9,2,10,6,4,8} 161 {1,2,6,4,10} 85 {1,9,2,10,6,7} 12{1,2,3,4} 48 {1,2,6,4,8,9,7} 162 {1,2,6,4,8} 86 {1,9,2,10,6,4} 12{1,2,3,7} 49 {1,2,6,4,8,9,3} 163 {1,2,6,3,4} 87 {1,2,6,4,10,8} 12{1,2,7,6} 50 {1,2,6,4,8,3,7} 164 {1,2,3,4,8} 88 {1,2,6,4,8,9} 13

{1,2,7,4} 51 {1,2,7,6,4,8,10} 165 {1,2,3,4,7} 89 {1,2,6,4,8,3} 13{1,2,3,7,8} 90 {1,2,3,4,7,8} 13{1,2,7,6,3} 91 {1,2,7,6,3,4} 13{1,2,7,6,4} 92 {1,2,7,6,4,8} 13{1,2,7,4,8} 93 {1,2,7,4,8,10} 13

Table 1: A generator. w(v) denotes the weight of the vertex v. In row 2 at number 186 the{1 − 10} denotes the 11-tuple {1, 3, 9, 4, 6, 5, 8, 2, 10, 7}.

17



# # φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C )C 1 Root C 2, C 1, 1, 186 C 3, C 1 1, 1861 {1} 2 {1,6} 5 {1,9}3 {1,3} 7 {1,6,3} 14 {1,9,5}9 {1,3,9} 23 {1,6,3,4} 40 {1,9,5,10}30 {1,3,9,4} 53 {1,6,3,4,7} 79 {1,9,5,10,2}

67 {1,3,9,4,6} 97 {1,6,3,4,7,5} 123 {1,9,5,10,2,6}114 {1,3,9,4,6,5} 139 {1,6,3,4,7,5,2} 157 {1,9,5,10,2,6,7}150 {1,3,9,4,6,5,8} 167 {1,6,3,4,7,5,2,8} 177 {1,9,5,10,2,6,7,4}173 {1,3,9,4,6,5,8,2} 181 {1,6,3,4,7,5,2,8,9} 184 {1,9,5,10,2,6,7,4,8}183 {1,3,9,4,6,5,8,2,10} # φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C )186 {1,3,9,4,6,5,8,2,10,7} C 4, C 1, 1, 186 C 5, C 1, 3, 186# φ(C ), φs(C ), φt(C ) 6 {1,2} 8 {1,3,8}C 6 C 1 3 183 19 {1,2,6} 25 {1,3,8,6}10 {1,3,5} 46 {1,2,6,4} 59 {1,3,8,6,2}35 {1,3,5,6} 86 {1,2,6,4,8} 104 {1,3,8,6,2,7}75 {1,3,5,6,10} 131 {1,2,6,4,8,3} 145 {1,3,8,6,2,7,9}120 {1,3,5,6,10,2} 164 {1,2,6,4,8,3,7} 170 {1,3,8,6,2,7,9,10}153 {1,3,5,6,10,2,4} 180 {1,2,6,4,8,3,7,10} 182 {1,3,8,6,2,7,9,10,4}175 {1,3,5,6,10,2,4,9} 185 {1,2,6,4,8,3,7,10,5} # φ(C ), φs(C ), φt(C )# φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C ) C 7 C 2 7, 181C 8 C 7 52, 166 C 9 C 7 52, 166 22 {1,6,3,7}94 {1,6,3,7,9,5} 96 {1,6,3,7,9,4} 52 {1,6,3,7,9}136 {1,6,3,7,9,5,2} 138 {1,6,3,7,9,4,5} 95 {1,6,3,7,9,2}# φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C ) 137 {1,6,3,7,9,2,4}C 10 C 2 7, 139 C 12 C 5 25, 182 166 {1,6,3,7,9,2,4,5}24 {1,6,3,10} 56 {1,3,8,6,10} # φ(C ), φs(C ), φt(C )

55 {1,6,3,10,7} 99 {1,3,8,6,10,7} C 11 C 10 24, 9898 {1,6,3,10,7,4} 141 {1,3,8,6,10,7,4} 54 {1,6,3,10,4}140 {1,6,3,10,7,4,5} 169 {1,3,8,6,10,7,4,9} # φ(C ), φs(C ), φt(C )168 {1,6,3,10,7,4,5,2} # φ(C ), φs(C ), φt(C ) C 13 C 5 59, 145# φ(C ), φs(C ), φt(C ) C 14 C 5 25, 170 103 {1,3,8,6,2,9}C 15 C 5 25, 170 58 {1,3,8,6,9} # φ(C ), φs(C ), φt(C )57 {1,3,8,6,7} 102 {1,3,8,6,9,10} C 16 C 15 57, 101101 {1,3,8,6,7,9} 144 {1,3,8,6,9,10,2} 100 {1,3,8,6,7,4}143 {1,3,8,6,7,9,10} # φ(C ), φs(C ), φt(C ) 142 {1,3,8,6,7,4,9}# φ(C ), φs(C ), φt(C ) C 18 C 5 8, 59 # φ(C ), φs(C ), φt(C )C 17

C 5

8, 59 27 {1,3,8,10} C 19

C 18

27, 6026 {1,3,8,2} 60 {1,3,8,10,2} 61 {1,3,8,10,9}

105 {1,3,8,10,2,6} 106 {1,3,8,10,9,2}

Table 2: φ(C ), φs(C ), and φt(C ) denote the parent of the chain C , the first element of the parent chain and the second element of the parent chain that connect the chain to theparent chain, respectively. The numbers, (bold faced or not), are the listing numbers of the11-tuples in Table 1.

18



# φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C )C 20 C 19 61, 106 C 21 C 1 9, 183 C 22 C 21 29, 10728 {1,3,8,9} 29 {1,3,9,2} 63 {1,3,9,2,10}62 {1,3,8,9,2} 64 {1,3,9,2,6} # φ(C ), φs(C ), φt(C )# φ(C ), φs(C ), φt(C ) 107 {1,3,9,2,6,10} C 23 C 21 64, 146C 24 C 1 30, 183 146 {1,3,9,2,6,10,4} 108 {1,3,9,2,6,4}

66 {1,3,9,4,8} 171 {1,3,9,2,6,10,4,8} # φ(C ), φs(C ), φt(C )110 {1,3,9,4,8,2} # φ(C ), φs(C ), φt(C ) C 25 C 24 110, 172148 {1,3,9,4,8,2,10} C 26 C 24 66, 148 65 {1,3,9,4,2}172 {1,3,9,4,8,2,10,5} 111 {1,3,9,4,8,10} 109 {1,3,9,4,2,10}# φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C ) 147 {1,3,9,4,2,10,5}C 27 C 24 66, 172 C 28 C 1 67, 150 # φ(C ), φs(C ), φt(C )112 {1,3,9,4,8,5} 113 {1,3,9,4,6,8} C 29 C 1 9, 114149 {1,3,9,4,8,5,10} # φ(C ), φs(C ), φt(C ) 31 {1,3,9,6}# φ(C ), φs(C ), φt(C ) C 30 C 1 9, 173 68 {1,3,9,6,5}C 31 C 30 70, 151 32 {1,3,9,5} # φ(C ), φs(C ), φt(C )33

{1,3,5,2

}70

{1,3,9,5,2

}C 32

C 31 33

,11772 {1,3,5,2,4} 116 {1,3,9,5,2,8} 71 {1,3,5,2,8}

117 {1,3,5,2,4,8} 151 {1,3,9,5,2,8,4} # φ(C ), φs(C ), φt(C )# φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C ) C 33 C 30 32, 151C 34 C 6 10, 153 C 35 C 34 34, 118 69 {1,3,9,5,4}34 {1,3,5,4} 73 {1,3,5,4,8} 115 {1,3,9,5,4,2}74 {1,3,5,4,6} # φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C )118 {1,3,5,4,6,8} C 36 C 6 75, 153 C 37 C 6 75, 120152 {1,3,5,4,6,8,10} 119 {1,3,5,6,10,4} 36 {1,5,6,10}174 {1,3,5,4,6,8,10,2} # φ(C ), φs(C ), φt(C ) 76 {1,5,6,10,2}# φ(C ), φs(C ), φt(C ) C 38 C 3 14, 123 # φ(C ), φs(C ), φt(C )

C 39 C 38 77, 122 4 {1,5} C 40 C 38 4, 122121 {1,5,6,2,3,4} 11 {1,5,6} 12 {1,5,2}154 {1,5,6,2,3,4,9} 37 {1,5,6,2} 38 {1,5,2,9}# φ(C ), φs(C ), φt(C ) 77 {1,5,6,2,3} 78 {1,5,2,9,6}C 41 C 3 40, 79 122 {1,5,6,2,3,9} # φ(C ), φs(C ), φt(C )13 {1,5,10} 155 {1,5,6,2,3,9,10} C 42 C 3 123, 18439 {1,5,10,2} # φ(C ), φs(C ), φt(C ) 156 {1,9,5,10,2,6,4}# φ(C ), φs(C ), φt(C ) C 43 C 3 5, 184 176 {1,9,5,10,2,6,4,8}C 45 C 43 43, 178 17 {1,9,2} # φ(C ), φs(C ), φt(C )16 {1,9,10} 43 {1,9,2,10} C 44 C 43 83, 17842 {1,9,10,6} 83 {1,9,2,10,6} 127 {1,9,2,10,6,7}82 {1,9,10,6,8,} 128 {1,9,2,10,6,4} 160 {1,9,2,10,6,7,4}126 {1,9,10,6,8,7} 161 {1,9,2,10,6,4,8}159 {1,9,10,6,8,7,2} 178 {1,9,2,10,6,4,8,7}

Table 3: φ(C ), φs(C ), and φt(C ) denote the parent of the chain C , the first element of the parent chain and the second element of the parent chain that connect the chain to theparent chain, respectively. The numbers, (bold faced or not), are the listing numbers of the11-tuples in Table 1.

19



# φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C )C 46 C 45 42, 126 C 47 C 45 42, 126 C 48 C 47 80, 15881 {1,9,10,6,7} 15 {1,9,6} 124 {1,9,6,5,10,8}# φ(C ), φs(C ), φt(C ) 41 {1,9,6,5} # φ(C ), φs(C ), φt(C )C 49 C 4 6, 19 80 {1,9,6,5,10} C 50 C 4 19, 4618 {1,2,10} 125 {1,9,6,5,10,7} 45 {1,2,6,9}44 {1,2,10,6} 158 {1,9,6,5,10,7,8} 84 {1,2,6,9,4}# φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C ) # φ(C ), φs(C ), φt(C )C 51 C 4 46, 86 C 52 C 4 86, 164 C 53 C 52 130, 179

85 {1,2,6,4,10} 130 {1,2,6,4,8,9} 163 {1,2,6,4,8,9,3}129 {1,2,6,4,10,8} 162 {1,2,6,4,8,9,7} # φ(C ), φs(C ), φt(C )# φ(C ), φs(C ), φt(C ) 179 {1,2,6,4,8,9,7,3} C 54 C 4 19, 131C 55 C 4 6, 164 # φ(C ), φs(C ), φt(C ) 47 {1,2,6,3}20 {1,2,3} C 56 C 55 48, 132 87 {1,2,6,3,4}48 {1,2,3,4} 88 {1,2,3,4,8} # φ(C ), φs(C ), φt(C )89 {1,2,3,4,7} # φ(C ), φs(C ), φt(C ) C 57 C 55 20, 132132 {1,2,3,4,7,8} C 58 C 4 6, 180 49 {1,2,3,7}# φ(C ), φs(C ), φt(C ) 21 {1,2,7} 90 {1,2,3,7,8}C 59 C 58 50, 92 50 {1,2,7,6} # φ(C ), φs(C ), φt(C )

91 {1,2,7,6,3} 92 {1,2,7,6,4} C 60 C 58 21, 165133 {1,2,7,6,3,4} 134 {1,2,7,6,4,8} 51 {1,2,7,4}165 {1,2,7,6,4,8,10} 93 {1,2,7,4,8}

135 {1,2,7,4,8,10}

Table 4: Tables 2, 3, and 4 show the vertices and edges of a chain-minimal saturated chaincover generator graph with 60 saturated chains. φ(C ), φs(C ), and φt(C ) denote the parentof the chain C , the first element of the parent chain and the second element of the parentchain that connect the chain to the parent chain, respectively. The numbers, (bold faced ornot), are the listing numbers of the 11-tuples in Table 1.

20



Figure 3: The chain cover tree with 60 saturated chains

21



3029

21 6

2 3 4

528 24

26 27

25

41

4243

38

Figure 4: Three conflict graphs of the chain cover tree of Figure 3

49 51 52

545558

5012 14 15

181713 34 36 37

Figure 5: Three conflict graphs of the chain cover tree of Figure 3

tree of Figure 3. They are all trivially bipartite and thus the extended chain cover graph isplanar. Figure 6 is a planar drawing of the extended chain cover graph.

Note that the extended chain cover graph is not a Venn model of a doodle.

Lemma 38. There is a minimal Venn model of the doodle obtained from the extended chain cover graph of Figure 6. This can be obtained by the procedure of Construction 24. Thisgraph has 57 faces.

Proof. The cover edges of the following chains are parallel: C 10, C 16, C 18, C 19, C 20, C 25, C 31,C 37, C 39, C 41, C 49, C 50, C 51, C 52 and C 59, while chain C 47 has reversed cover edges, finallychain C 45 has half reversed cover edges. The three, two, four, and two chains in the follow-ing four sets of chains {C 18, C 19, C 20}, {C 37, C 39}, {C 49, C 50, C 51, C 52}, and {C 45, C 47} violateproperty 2. of Definition 20 in one single but in four different faces. All the other chainslisted here violates property 2. of Definition 20 in other different faces. Furthermore, chainsC 18, C 19, C 20, and C 45, C 47 are connected with cover edges, see Figure 3. Adding 16 crossingedges one can create a doodle of a Venn model. Table 5 shows the crossing edges. In thetable the crossing edges are represented by a pair of numbers; these numbers are from Table 1.

It is easy to check that by adding these crossing edges we create a Venn model of an 11-doodle, see Figure 7. To create a minimal Venn model of a 11-doodle the following 20 edgescan be removed without violating Definition 20. Table 6 lists the set of removable edges.The resulting graph is in Figure 8

22



Figure 6: The extended chain cover graph

23



Figure 7: The extended chain cover graph with the added crossing edges

24



{98, 141} {142, 169} {62, 29} {106, 63}{30, 65} {10, 33} {174, 183} {37, 76}{154, 175} {4, 13} {5, 15} {44, 83}{84, 128} {85, 128} {130, 161} {49, 91}

Table 5: A list of the crossing edges

{101, 142} {25, 59} {27, 60} {62, 106}{29, 63} {64, 107} {65, 110} {148, 172}{36, 120} {77, 122} {13, 40} {15, 42}{128, 161} {19, 44} {46, 84} {46, 85}{86, 131} {50, 91} {75, 120} {67, 114}

Table 6: A list of the removable edges

The obtained minimal Venn model of a 11-doodle has 57 feces and thus the non-simple 11-doily created from this doodle has 11 × 57 = 627 vertices. Indeed, the 60 chains create 61faces. The added 16 crossing edges add another 16 faces, this is altogether 77 faces. Theremoved 20 edges lower this number to 57 faces. The minimal Venn model of the 11-doodlewith the edge numbers is shown in Figure 6.

The 11-doily with the vertex set of size 627 is in Figure 10. The intersection of the interiorof one of the Jordan curves and the 11-doily can be seen in Figure 11. Finally the Jordancurve that creates this doily is in Figure 12. All of the figures of a Venn model of a doily (ora Venn model of a doodle) have some vertices that are incident to edges that have only oneend, with the understanding that in the exterior they are all incident with the single vertex

< 0, 0, . . . , 0 >.

25



Figure 8: A minimal Venn model of a doodle of a non-simple 11-doily with 627 vertices andwith the edge numbers. The edge numbers are from Table 1.

26



Figure 9: A minimal Venn model of the 11-doily with the vertex set of size 627 .

27





Figure 11: The intersection of the interior of one of the Jordan curves and the 11-doily fromFigure 10. The outer contour of this figure is one of the Jordan curves in the non-simple11-doily with 627 vertices.

29



Figure 12: The Jordan curve, and the center of rotation. The rotation of this curve 11 timesover 360/11 degrees creates the non-simple 11-doily with 627 vertices in Figure 10.

30



3 A maximal Venn model and the 11-doily with 1837

vertices

Note that the graph of this doodle is extendable. The vertices of the Venn model of thedoodle which can be connected without violating Definition 20 can be seen in Figure 13. Inaddition, all of the eleven pairs of vertices can be connected between two consecutive copiesof the doodle. Figure 13 also shows all the edge numbers created by these new edges. InFigure 13 there are 110 new pairs of vertices, that is 110 new edges. This is the maximumnumber of new edges that can be added to this model of the doodle without violating Defi-nition 20.

These edges can be added one-by-one. Also one can add any set of these edges to the modelof the doodle. 110 steps, each adding one new edge, to create 110 different 11-doilies withvertex sets 627, 627 + 11, . . . , 1837. Adding any subset of this edge set to the model of thedoodle creates 2110 different apparently non-isomorphic models. Combining the extensionwith deletion of edges may provide even more new models, we did not however explore this

road.

With limited space for publication we only show here the maximal Venn model of the doodlein which all of the possible pairs of vertices are connected without violating Definition 20.This is in Figure 13. Figure 14 shows the Venn model of the doily. Figure 15 shows the11-doily which is created by this Venn model. Figure 16 shows the intersection of the interiorof one of the Jordan curves and the 11-doily in Figure15. Figure 17 shows the Jordan curve.The rotation of this curve over 360/11 degrees creates the non-simple 11-doily with 352vertices.In Figures 10 and 15 there are 2048 faces, 627 or 1837 vertices, and 2076 or 3883 edges, re-spectively. The difficulty is to show a decent drawing of such diagrams in a paper of limited

size.

4 Problems and conjectures

Conjecture 39. For every integer n ≥ 2 and for every k ∈ Kn there is a SaCP C of theBoolean lattice Bn such that |C| = k, (see Kn in Notation 21).

Problem 40. Is it possible to construct the SaCP in Conjecture 39 with the chain cover property?

Remark 41. Note that the SaCP following from our construction is not unique, some havethe chain cover property, some others do not.

Problem 42. For every prime number p and for any m ∈ Λ p there is a chain-minimal SaCCGG G such that |G(L)| = m, (see Λ p in Notation 25).

Conjecture 43. For every prime number p there is a fully extendable p-Venn model. Isthere a chain-minimal SaCCGG, such that the conjectured model is corresponding to it?

31



Figure 13: A maximal Venn model of a doodle of a non-simple 11-doily with 1837 vertices.The new edges added to the Venn model of the doodle of Figure 6. The edge numbers arefrom Table 1.

32



Figure 14: A maximal Venn model of a non-simple 11-doily with 1837 vertices.

33



Figure 15: A non-simple 11-doily with 1837 vertices.

34



Figure 16: The intersection of the interior of one of the Jordan curves and the 11-doily fromFigure 15. The outer contour of this figure is one of the Jordan curves in the non-simple11-doily with 1837 vertices.

35



Figure 17: The Jordan curve, and the center of rotation. The rotation of this curve 11 timesover 360/11 degrees creates the non-simple 11-doily with 1837 vertices in Figure 15.

36



Problem 44. For every prime number p there is a λ ∈ Λ p and a SaCCGG G, |G(L)| = λ,such that the corresponding p-Venn model is fully extendable.

Problem 45. For every prime number p and for every k ∈ K p is there a minimum chain-minimal p-Venn model with k-many saturated chains which is fully extendable?

The statements of Conjecture 42, Problems 44, and 45 look redundant. In fact it is not true

that each Venn model of a doily can be constructed by a Venn model of a doodle! It is nothard to check that some of the 7-doilies of [17] cannot be constructed this way.

Problem 46. Is there a minimum chain-minimal p-Venn model G which is generated by a SCCGG G∗, and which is fully extendable? In particular, is it possible to modify theconstruction of [6] to obtain such SCCGG for any prime number p > 7?

Remark 47. For prime numbers p = 2, 3, 5, and 7 the requested SCCGG in Problem 46 exists, this is shown in [11].

Problem 48. Find either a necessary, or a sufficient, or a necessary and sufficient condition to correct the violation of property 2. of Definition 20 in a planar extended chain cover graph.

5 Acknowledgment

The authors thank Professor Adam Coffman and the referees for their valuable suggestions.The figures were created by artist Edit Hepp; they are her intellectual properties; and shehas all the copyrights. The authors are deeply thankful to her for letting these marvelousimages be used in this paper.

References

[1] B. Bultena, B. Grunbaum, and F. Ruskey, Convex drawings of intersecting fam-ilies of simple closed curves, (preprint) (1998).

[2] N. G. de Bruijn, C. Tengbergen, and D. Kruyswijk, On the set of divisors of a number, Nieuw. Arch. Wiskunde (2) 23 (1951), 191–193.

[3] K. B. Chilakamarri, P. Hamburger and R. E. Pippert, Venn diagrams andplanar graphs, Geometriae Dedicata 62 (1996), 73–91.

[4] A. W. F. Edwards, Seven-set Venn diagrams with rotational and polar symmetry,Combinatorics, Probability, and Computing , 7 (1998), 149–152.

[5] C. Green and D. J. Kleitman, Strong version of Sperner’s theorem, Journal of Combinatorial Theory Ser. A 20/1 (1976), 80–88.

[6] J. Griggs, C. E. Killian, and C. D. Savage, Venn diagrams and symmetric chaindecompositions in the Boolean lattice, (preprint).

[7] B. Grunbaum, Venn diagrams I, Geombinatorics 1 (1992), 5–12.

37



[8] B. Grunbaum, Venn diagrams II, Geombinatorics 2 (1992), 25–32.

[9] B. Grunbaum, The search for symmetric Venn diagrams, Geombinatorics 8 (1999),104–109.

[10] P. Hamburger, Doilies and doodles, Discrete Mathematics 257/2-3 (2002), 423–439,a Special Issue in Honor of the 65th Birthday of Daniel J. Kleitman.

[11] P. Hamburger, Pretty drawings. More doodles and doilies, symmetric Venn dia-grams, Utilitas Mathematica, (in print).

[12] P. Hamburger and A. Sali, Symmetric 11-Venn diagrams with vertex sets231, 242, . . . , 352, Studia Mathematica Hungarica , 40/1-2 (2003), 121–143.

[13] P. Hamburger and A. Sali, 11-Doilies with Vertex Sets 275, 286, . . . 462, (submit-ted).

[14] P. Hamburger, Assigning binary bit string codes to planar regions using symmetric

Venn diagrams, (in preparation).[15] D. W. Henderson, Venn diagrams for more than four classes, Mathematics Magazine

4 (1963), 424–426.

[16] A. Renyi, K. Renyi, and J. Suranyi, Sur l’independance des domaines simplesdans l’espace Euclidien a n-dimensions, Colloquium Mathematicum 2 (1951), 130–135.

[17] F. Ruskey, A survey of Venn diagrams, The Electronic Journal of Combinatorics, 4(1997), DS#5. see also: http://sue.csc.uvic.ca/˜ cos/venn/VennEJC.html.

[18] A. J. Schwenk, Venn diagram for five curves, Mathematics Magazine 57 (1984), 297.

[19] R. Stanley, Enumerative Combinatorics, Cambridge University Press, Volume 1,(1997), and Volume 2, 1999.

Documents

Peter Hamburger, Gyorgy Petruska and Attila Sali- Saturated Chain Partitions in Ranked Partially Ordered Sets, and Non-Monotone Symmetric 11-Venn Diagrams