Geometrical constructions of class-uniformly resolvable structure

Geometrical Constructions ofClass-Uniformly Resolvable Structures

Peter Danziger,1 Malcolm Greig,2 Brett Stevens3

1Department of Mathematics, Ryerson University, Toronto, Canada ON M5B 2K3,E-mail: [email protected]

2Greig Consulting, North Vancouver, BC, Canada, E-mail: [email protected]

3School of Mathematics and Statistics, Carleton University, Ottawa, Canada ONK1S 5B6, E-mail: [email protected]

Received October 26, 2009; revised February 11, 2011

Published online 16 June 2011 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/jcd.20284

Abstract: We use arcs, ovals, and hyperovals to construct class-uniformly resolvable struc-tures. Many of the structures come from finite geometries, but we also use arcs from non-geometric designs. Most of the class-uniformly resolvable structures constructed here haveblock size sets that have not been constructed before. We construct CURDs with a varietyof block sizes, including many with block sizes 2 and 4. In addition, these constructions givethe first systematic way of constructing infinite families of CURDs with three block sizes.q 2011 Wiley Periodicals, Inc. J Combin Designs 19: 329–344, 2011

Keywords: class-uniformly resolvable design; finite geometry; oval; subplane; unital

1. INTRODUCTION

A Pairwise Balanced Design, PBD(v,K ,�), is a pair (X,B) where |X |=v and B is acollection of subsets of X , called blocks. Each subset has size k∈K and each pair ofpoints of X occurs exactly � times in the blocks. If there is only one block size, k, thenit is called a Balanced Incomplete Block Design and we write BIBD(v,k,�).A Group Divisible Design, (K ,�)-GDD of type

∏gugii , is a triple (X,G,B) where

|X |=v=∑i gi ugi and G is a partition of X into groups, for each G∈G |G|=gi for

Contract grant sponsor: NSERC (to P. D. and B. S.).

Journal of Combinatorial Designsq 2011 Wiley Periodicals, Inc. 329

330 DANZIGER, GREIG, AND STEVENS

some i and there are precisely ugi groups of cardinality gi . B is a collection of subsetsof X , called blocks. Each block has size k∈K and each pair of points of X occurs inexactly � blocks or one group, but not both. If there is only one block size, k, then it iscalled a (k,�)-GDD of type

∏gugii . If all the groups are of the same size, g, and k=u,

then this is a Transversal Design, TD�(k,g).If the blocks of a design can be partitioned into resolution classes such that each

resolution class contains each member of X exactly once, then the design is calledresolvable and we write RPBD(v,K ,�), RBIBD(v,k,�), K -RGDD�, or RTD�(k,g) asappropriate.We will assume that the reader is acquainted with other design theory terminology

and we refer them to [4] for further definitions and notation. We also assume that thereader is familiar with the basic properties of finite geometries, see for example [8]. Inparticular, in PG(2,q), there are q2+q+1 points and the same number of blocks, everyline has size q+1 and every point is incident with q+1 lines, and every pair of linesintersects in exactly one point; in AG(2,q), there are q2 points, q2+q blocks, everyline has size q , every point is incident with q+1 blocks and for every point P and line,� not containing P , there is a unique line containing P which is disjoint from (parallelto) �. The relation between geometries and designs has a long history, in particular aPG(2,q) is a BIBD(q2+q+1,q+1,1) and an AG(2,q) is a RBIBD(q2,q,1). Further,a RTD(n,n) is equivalent to a TD(n+1,n) and both are equivalent to a Projective PlanePG(2,n); the only known values when such designs exist are when n is a prime power.If every resolution class of a resolvable design has the same number of blocks of

each size, we say that the resolution is class-uniform, and we call the design a Class-Uniformly Resolvable Design (CURD) or Class-Uniformly Resolvable Group DivisibleDesign (CURGDD) as appropriate. More specifically given a resolvable design with rresolution classes and K ={k1, . . .kn}, ki<ki+1, then for each ki there is a correspondingpki such that

∑ni=1 pki ki =v and each resolution class contains exactly pki blocks

of size ki . We describe the resolution classes in exponential notation calling thema partition k

pk11 · · ·k pknn . In this article we concentrate on the case where �=1 and

hence omit this parameter. CURDs and CURGDDs have been previously investigatedin [5–7, 15, 16, 20].The general problem of finding CURDs and CURGDDs is extremely difficult.

A well-known variant of the problem is where the blocks are replaced with cycles; inthis case a class-uniform structure becomes a solution to the Oberwolfach problem,which was posed by Ringel in the 1960s. Cycle decompositions are typically easierthan block designs; for example, the complete necessary and sufficient conditionsfor fixed size cycle decompositions are known [1, 17]. However, the class-uniformresolvable problem, even when restricted to the easier realm of cycles is notoriouslydifficult, currently only solved in absolute generality for any pair of odd cycle sizes[14] or for any collection of even cycle sizes [3]. It is also solved in a number of othermore specific instances [4].Due to the difficulty of the CURD problem, most of the results thus far in the literature

focus on the case of block sizes 2 and 3 (or 1 and k) as these have been the onlyapproachable values. This does not mean that the investigation of other block sizes isuninteresting, rather that even the case of sizes 2 and 3 is hard and is far from solved.In this article, we create infinite families giving other values for the block sizes, relatedto powers of primes. In particular, these can be used to create designs with block sizes2 and 4. Though some CURGDDs with block sizes 2 and 4 appear in [6], we believe

Journal of Combinatorial Designs DOI 10.1002/jcd

CLASS-UNIFORMLY RESOLVABLE STRUCTURES 331

that our is the first construction of infinite families of CURDs with these block sizes.We also present constructions which work for many other block sizes related to powersof primes.Most attempts to construct CURDs and CURGDDs have been either recursive or

direct from difference methods. In this article, we construct class-uniform structuresusing the substructure of finite geometries and designs. Using these methods allows usto consider block sizes which were previously unattainable, mostly values related topowers of primes.In [12], Greig introduced the idea of using the arcs and ovals of a geometry to create

designs. In this article, we extend these ideas and use such constructions to produceclass-uniform structures. In Section 2, we review the basic geometric objects: arcs,ovals, and subplanes, which we then use in the constructions of Section 3.

2. ARCS, OVALS, AND SUBPLANES

In this section, we introduce arcs and present some known families.

Definition 2.1. Given a set S⊆N and a design with point set V , an S-arc is a set,W⊆V , of w points such that every block in the design intersects W in s points forsome s∈ S.

We call a line that intersects the arc in i points an i-line; a 1-line is called a tangent.The complement of an S-arc on w points in a BIBD(v,k,�) is a T -arc on v−w points,where T ={k−s|s∈ S}. The intersection parameters of an arc are the number of i-linesthe various points of the design are incident with. There may often be multiple types ofpoints with respect to these parameters and in such cases, we will present the intersectionparameters in a table; when there are only a few simple types of points, we will simplystate the parameters in the text.A {0,1,2}-arc in PG(2,q) that is of maximum cardinality, q+1, is called an oval.

Each point on the oval is incident with a unique tangent and exactly q 2-lines. Thenumber of i-lines through points external to the arc is not fixed, but there are onlytwo types of points. The kinds and number of intersections between lines and pointsexternal to the oval are given in Table I. When q is even, all q+1 tangents intersect ina common point which we call the nucleus. Inclusion of the nucleus in the oval resultsin a {0,2}-arc of cardinality q+2 called a hyperoval.Denniston has shown the existence of {0,2n}-arcs in PG(2,2N ) [9]. Greig has pointed

out that since his construction makes use of subgroups which may be nested, we getarcs that may also be nested [12].Theorem 2.2 (Denniston [9] and Greig [12]). Let 0≤n≤N , and let vN (x)=(2x −1)2N +2x , then PG(2,2N ) contains a {0,2n}-arc on vN (n) points. Furthermore, if m<n,

then there is a subset of size vN (m) of those vN (n) points that is a {0,2m}-arc.We call these arcs Denniston arcs. By deleting the points of the smaller Denniston arcfrom the geometry, we get:

Corollary 2.3 (Denniston [9] and Greig [12]). If 0≤m<n≤N , there is a {0,2n,2n−2m}-arc on (2N +1)(2n−2m) points in PG(2,2N ).



TABLE I. Intersection Parameters of Points Relative to an Oval in PG(2,q).

Number of lines eachpoint is incident with

Parity Type Numberof q (external to Oval) of points 0-lines 1-lines 2-lines

Inside Any – q+1 0 1 q

Outside q odd Type 1q(q−1)

2

q+1

20

q+1

2the Type 2

q(q+1)

2

q−1

22

q−1

2Oval q even Type 1 1 0 q+1 0

Type 2 q2−1q

21

q

2

The number of nested arcs is not limited to two and we will make use of a series of suchnested arcs in Section 3B. The nesting properties described in Corollary 2.3 hold ingeneral, but we wait until that section to work out the specific intersection parameters.Because there is always a line disjoint from a Denniston arc, we can consider that to

be the line at infinity in the projective plane and thus find these arcs embedded in affinegeometries and utilize the resolvability:

Corollary 2.4 (Denniston [9] and Greig [12]). Let 0≤n≤N , and let vN (x)=(2x−1)2N +2x . Then AG(2,2N ) contains a {0,2n}-arc on vN (n) points. Furthermore, if m<n,

then there is a subset of size vN (m) of those vN (n) points that is a {0,2m}-arc.Greig and Rosa have found another corollary of Theorem 2.2. For any a∈GF(2n)\{0},

{0,a} forms an additive subgroup of GF(2n) isomorphic to GF(2). Using this theyobtain:

Corollary 2.5 (Greig and Rosa [13]). There is a point in AG(2,2n) which is thecommon point of 2n−1 hyperovals such that every other point of AG(2,2n) lies in justone of these hyperovals.

Baer has constructed a family of subplanes, which are also S-arcs:

Theorem 2.6 (Beth et al. [2]). When q is a prime power, and b|a then there existssubplane PG(2,qb) in PG(2,qa). This subplane is a {0,1,qb+1}-arc on q2b+qb+1points. If a=2b, then this arc is a {1,qb+1}-arc and is called a Baer subplane.

The intersection parameters for these arcs are given in Table II.A unital is a BIBD(n3+1,n+1,1) and many are embeddable in projective planes as

arcs.

Theorem 2.7 (Fisher et al. [10]). If q is a prime power, then there exists a BIBD(q3+1,q+1,1) embeddable in PG(2,q2) as a {1,q+1}-arc.Every point on the embedded unital is on q2 (q+1)-lines and exactly one tangent.Every point that is external to the embedded unital is on (q2−q) (q+1)-lines and q+1tangents.Rosa and Greig have constructed the following S-arcs in RBIBDs.



TABLE II. Intersection Parameters of Points, Lines, and Subplanes of PG(2,qa).

Number of lines eachpoint is incident with

Location Points Numberof points of plane of points 0-lines 1-lines (qb+1)-lines

Inside the q2b+qb+1 0 qa−qb qb+1arc

Outside Type 1 (q2b+qb+1)(qa−qb) qa−q2b q2b 1The arc Type 2 (qa−q2b)(qa+qb) qa−q2b−qb q2b+qb+1 0

Theorem 2.8 (Greig and Rosa [13]). For every t≥0, there exists a RBIBD(12t+4,4,1) which consists of three {0,2}-arcs on 4t+2 points, all intersecting in a commonpoint.

More generally, Greig and Rosa have pointed out that several difference familiesconstructed by Greig [11] have similar properties.

Theorem 2.9 (Greig [11] and Greig and Rosa [13]). If q=kt+1 is a prime power,there exists a RBIBD((k−1)q+1,k,1) which consists of k−1 maximal {0,2}-arcs (ofsize q+1), all intersecting in a common point in the following circumstances:1. For k=4, all t;2. for k=6, t even, t �∈ {2,6}, and t≤832;3. for k=8, t �∈ {3,11} and t≤512;4. for k=10, t even, t �∈ {4,6,10,12,18,24}, and t≤729.

The intersection parameters for both Theorem 2.8 and 2.9 are the same when statedin terms of k and t . There are kt+2 internal points, each is on q 2-lines and zero 0-lines.There are (k−2)(kt+1) points external to each arc which are on (kt+2)/2 2-lines andkt/2 0-lines.

3. CONSTRUCTIONS

In this section, we give constructions for class-uniform objects using arcs in geometriesand designs. Typically, the point sets of the constructed designs are the points of an arcor its complement. Usually, we derive the resolution classes and groups using the pointsof a line; each resolution class consists of the set of lines through one of those pointsrestricted to the arc. We may also inherit resolvability from the larger design when thatis possible. In either case, the uniformity of the way in which these classes intersect thearc yields the class-uniformity.We also will need the following straightforward residue design construction.

Theorem 3.1. If a CURD� with partition kpk11 . . . (k j −1)pk j−1k

pk jj . . .k

pknn exists and

contains some point that only occurs on blocks of size k j , then there exists a CURD�

with partition kpk11 . . . (k j −1)pk j−1+1k

pk j −1

j . . .kpknn .

Proof. Remove the specified point from the design. �



There is a corresponding point addition theorem.

Theorem 3.2. If a CURD� with partition kpk11 . . .k

pk jj (k j +1)pk j+1 . . .k

pknn exists and

each resolution class contains a block of size k j such that the set of these blocks form

a strength 1�-design, then there exists a CURD� with partition kpk11 . . . (k j )

pk j −1(k j +

1)pk j+1+1. . .k

pknn .

Proof. Add a new point and put it on each of the chosen blocks of size k j . �

A. Oval Constructions

We begin with arc constructions that make use of ovals and hyperovals in finite projectiveplanes. For reference, we give the various intersection parameters in Table I.Our first arc construction is quite typical of these methods.

Theorem 3.3. When q is an odd prime power, there exists a CURGDD of type (q−1)q ,r =q, and partition (q−2)(q−1)/2(q−1)1q(q−1)/2.

Proof. Remove an oval, O and a line � tangent to it from PG(2,q). The point ofintersection of � to the ovalO, P say, defines the groups. Each other point of the tangentline is used to define a resolution class.Every line through P other than � intersects the oval in exactly two points and

therefore yields q groups of size q−1. The q lines through each other point of thechosen tangent line are exactly:

• one other tangent line yielding a block of size q−1;• (q−1)/2 0-lines yielding blocks of size q , and• (q−1)/2 2-lines yielding blocks of size q−2. �

The previous construction is related to the powerful line-flip method outlined in [12].In a line-flip, one line is chosen and its points of intersection with the arc are removedand the remaining points of the line are added, yielding another arc with a differentstructure.A similar construction can be obtained by removing one point from the oval and

using that point to index the groups and all other points on the unique tangent at thatpoint to index resolution classes. However, this yields groups of size one and partition112(q−1)/2, which is just a near 1-factorization of Kq .

Theorem 3.4. For all odd prime powers q, there exists a CURGDD of type ((q−1)/2)q , r =q, and partition ((q−1)/2)(q−1)/2((q+1)/2)(q−1)/2.

Proof. Choose an oval O in PG(2,q) and remove the points of all tangents to the oval.This leaves only the q(q−1)/2 points of PG(2,q) external to the oval which do notlie on any tangent to it (see Table I). We choose one of the removed tangent lines, �

say, to define the groups and the resolution classes in a similar manner to the proof ofTheorem 3.3.Let P be the point of incidence of � on O. Since there is only one tangent through

any point, besides � there are additional q lines through P , each of which is a 2-line andso intersects the arc in exactly one other point. Consider one of these 2-lines throughP , �′ say, and let R be the other point of intersection of �′ with O. Now consider the



intersection of the tangents with �′. One tangent will intersect �′ at R and the rest mustintersect �′ at points external to O. But since each point on a tangent line is on twotangent lines, they are used in pairs and we have (q−1)/2 points remaining on �′.We will now establish the partition type of this CURGDD. Consider a point Q on �

other than P , which will have additional q lines through it. Since Q is incident with atangent line (namely �), it must be of type 2; so one other line through it is tangent andall of its points will be removed. There are (q−1)/2 lines through Q that do not intersectO at all, and each one of these must still intersect with the remaining q−1 tangent linesand they must do so in pairs, leaving (q+1)/2 points on each of these lines.There are also (q−1)/2 2-lines through Q; now let �′ be such a 2-line through Q.

Let R and R′ be the two points of �′ which lie on O. Again we consider the intersectionof �′ with the tangents of O, two of them intersect �′ at R and R′ and the rest mustintersect �′ in pairs, leaving (q−1)/2 points on �′. This establishes that the partition is(

q−1

2

)(q−1)/2(q+1

2

)(q−1)/2

.

�

Similar constructions, keeping all points that are on no tangents and additionallykeeping all but one of the points of the oval, or keeping all points that lie on q ofthe tangents, with or without the corresponding points of the oval give the followingtheorem.

Theorem 3.5. Let q be an odd prime power, then there exists:1. a CURGDD of type ((q+1)/2)q , replication number r =q,

and partition 11((q+1)/2)(q−1)/2((q+3)/2)(q−1)/2;2. a CURGDD of type ((q−1)/2)q , replication number r =q,

and partition (q−1)1((q−3)/2)(q−1)/2((q−1)/2)(q−1)/2;3. a CURGDD of type ((q+1)/2)q , replication number r =q,

and partition q1((q+1)/2)(q−1)/2((q−1)/2)(q−1)/2.

Proof. In each case we start with an oval O in PG(2,q), a tangent line �, and the pointof intersection of the tangent � with O, P . As in the proof of Theorem 3.4 above, weuse the pencil of lines through P to define the groups and the pencil through each pointQ on � to define the blocks. Each construction differs in the points of PG(2,q) whichare kept and left.

1. We proceed as in Theorem 3.4, except that we also retain all the points of theoval other than the point P . All remaining lines through P are 2-lines and socontain one extra point (the point other than P in the oval); thus, the groups areone point larger. When considering points Q on � to define the parallel classes,one of the lines through Q is a tangent line and all points on it, except where itintersects O, will be removed; this gives a block of size 1. The 0-lines through Qremain unchanged. The 2-lines through Q will pick up the two points in O, andso will be of size (q−1)/2+2=(q+3)/2.

2. In this case, we keep only those points which lie on tangents to O, other than �,but not in O and proceed in a similar manner to Theorem 3.4. When consideringthe lines through P to define the groups, the count proceeds as in Theorem 3.4,



except that now we keep the points on tangent lines instead of dropping them,leaving groups of size (q−1)/2.Again we define the parallel classes by considering lines through the other pointsQ of �. Since Q lies on a tangent line (namely �), it is of type 2 and so there isexactly one other tangent line through Q, which yields a block of size q−1, afterremoving Q and the point of intersection withO. Q is also incident with (q−1)/20-lines, each of which will intersect the q−1 tangent lines in pairs, leaving blocksof size (q−1)/2. Finally, Q is incident with (q−1)/2 2-lines; we remove the 2points in O and Q itself, each of which intersect one tangent line other than �, theremaining q−3 tangent lines intersect the block in pairs, leaving (q−3)/2 points.

3. This case is similar to case 2 above, except that now we also keep the points in O,other than P , as well. The groups now all gain the other point in O of the 2-linesthrough P . The tangent line through Q keeps the point of intersection with O andso has size q . The 0-lines are unaffected, but the 2-lines each gain the 2 points ofintersection with O, giving them size (q+1)/2. �

The corresponding constructions for even q yield the following results.

Theorem 3.6. For all even prime powers q, there exists a CURD with v=q2−q−1,r =q+1, and partition (q−2)q/2(q−1)1q(q−2)/2.

Proof. Remove an oval and a line disjoint from the oval from PG(2,q). The points onthe line are all of Type II and we use the lines through each to define a resolution class.Since each of these lines intersects the oval in 0, 1, or 2 points, we obtain the givenpartition (see Table I). �

Using Theorem 3.6 and removing the point of tangency, which lies only on tangentsto the oval and therefore satisfies the hypothesis of Theorem 3.1, gives the followingCorollary.

Corollary 3.7. For all even prime powers q, there exists a CURD with v=q2−q−2,r =q+1, and partition (q−2)(q+2)/2q(q−2)/2.

We could have proven this theorem directly by removing a hyperoval and a line disjointfrom it.The following is a modified CURGDD which has two orthogonal sets of groups; see

[19] for further discussion of modified structures.

Theorem 3.8. For all even prime powers q, there exists a modified CURGDD onv=q2−q points with two orthogonal classes of groups each of type (q−1)q and q−1resolution classes with partition (q−2)q/2qq/2.

Proof. Remove a hyperoval and a 2-line, �, through it from PG(2,q). The points ofintersection of � with the hyperoval each define a group. The remaining points of � eachdefine a class. �B. Denniston Arc Constructions

We now apply similar methods to the nested Denniston arcs from Section 2. First, wederive the required intersection parameters. Let V be the point set of an AG(2,2N ). LetvN (x)=2N (2x −1)+2x , let N = i0≥ i1> · · ·>il ≥0. By Denniston’s Corollary 2.4 and



Greig’s observation [12] about the nestability of Denniston Arcs, there are a series ofnested arcs with point sets

Vil ⊂Vil−1 ⊂·· ·Vi2 ⊂Vi1 ⊂V

with |Vi j |=vN (i j )=2N (2i j −1)+2i j . We will need to count the numbers of lines whichintersect subsequences of these arcs. To this end, we define b0 as the number of linesmissing Vi1 , b1 as the number of lines intersecting Vi1 (in 2i1 points), b1 j−10 as thenumber of lines hitting Vi j−1 and missing Vi j , and b1 j−11 as the number of lines hittingVi j for 1≤ j<l. Because the arcs are nested if a line intersects Vi j , it must intersectVi j ′ for all j ′< j . Similarly if a line misses Vi j , then it must miss Vi j ′ for all j ′> j . Byconsidering that every pair of points in Vi j must lie on a line and that this line mustintersect Vi j in precisely 2i j points and the fact that b1 j−10+b1 j−11=b1 j−1 , we get

b1 j = (2N +1)(2N −2N−i j +1)

b1 j−10 = (2N +1)(2N−i j −2N−i j−1)

To determine the number of lines from each resolution class of the AG(2,2N ), we do asimilar calculation counting the number of lines through each of the points on the lineat infinity from PG(2,2N ). This determines that the number of lines of each type fromeach resolution class is simply the total number divided by 2N +1. We know that if aline intersects Vi j non-trivially, then it intersects it in exactly 2i j points. Similarly, wecan determine the number of points that any given line intersects each of the Vi j \Vi j+1 .If a line, �, hits Vi j , then one of two cases occurs. If it also hits Vi j+1 , then

|�∩(Vi j \Vi j+1)|=2i j −2i j+1

Alternatively, if it misses Vi j+1 then

|�∩(Vi j \Vi j+1)|=2i j

To use these nested arcs to construct CURDs, we will alternate inclusion and exclu-sion of the points from the arcs. The only difference between this construction andthose of Section 3A is the increased complexity of calculating the parameters of theresulting CURDs. In all cases, the CURD inherits its resolution classes from the under-lying affine plane.

Theorem 3.9. Let vN (x)=2N (2x −1)+2x , let N = i0≥ i1> · · ·>il ≥0. Define

v =t=l∑t=1

(−1)t+1vN (it ) (1)

k j =

⎧⎪⎨⎪⎩0 for j =0t= j∑t=1

(−1)t+12it for 0< j ≤ l(2)

e j ={b1 j0/(2

N +1)=(2N−i j+1 −2N−i j ) for 0≤ j<l

b1l/(2N +1)=(2N −2N−il +1) for j = l

(3)



Then there is a CURD with partition ke11 . . .kell on v points, and a CURD with partition(2N −k0)e0 . . . (2N −kl)el on 22N −v points.If l is odd, then there exists a CURDwith partition ke11 . . .kel−1

l (kl −1)1 on v−1 points.If l is even, then there exists a CURD with partition (2N −k0)e0 . . . (2N −kl)el−1(2N −kl −1)1 on 22N −v−1 points.All of these CURDs have r =2N +1.

Proof. We consider the nested arcs on vN (i j ) points, Vi j , for 0< j ≤ l. The first CURDis constructed by including the points of arc Vi1 , excluding the points of Vi2 , but includingthe points of Vi3 and so on, giving

V=l⋃

j=1j odd

(Vi j \Vi j+1)

with the convention that if l≡1 mod 2, then Vil+1 =∅. The derivation of the size, v, ofthis set in Equation (1) should now be clear.To calculate the block sizes in the CURD, we must consider the previously calculated

sizes of intersections between the lines and each Vi j \Vi j+1 . For any given line, we onlyneed to consider the nested arcs that it intersects and thus can ignore all the arcs afterthe first one it misses. The calculation of ki in Equation (2) now gives the resultingblock size for a line that intersects Vi j and misses Vi j+1 .Equation (3) now derives the number of blocks of each size, which for e j is simply

the number of blocks per resolution class that intersect Vi j and miss Vi j+1 , except whenj = l where it is the number of blocks from each resolution class of AG(2,2N ) thathit Vil .The resolutions for the CURD are simply inherited from the resolution classes of the

AG(2,2N ) and thus there is the same number, r =2N +1.The second CURD is the complement of the first in AG(2,2N ).Note that if l is odd, then all vN (il) points of the smallest arc are included in the

first CURD and lie only on blocks of size kl . Similarly, if l is even then all vN (il)points of the smallest arc are included in the second CURD and lie only on blocks ofsize 2N −kl . In either case, we can use Theorem 3.1 to construct the third and fourthCURDS. �

Example 3.10. In AG(2,32), we have nested arcs on 1024, 496, 232, 100, 34 points.With v=1024−496+232−100+34=694 points, we have a CURD with partition3211622442082217; its complement (in AG(2,32)) is a CURD on 330 points with apartition of 162841281017. In the first CURD, all 34 of the points of the smallest arc lieonly on blocks of size 22; thus, we can use Theorem 3.1 to get a CURD with partition3211622442082216211.

A few of the more simple corollaries of Theorem 3.9 that yield CURDs with a smallnumber of block sizes are worth stating explicitly. By setting l=1 and i1=n, we get:

Corollary 3.11. Let q=2N and let n≤N. There exists a CURDwith v=2n(2N−n−1)(2N +1), r =2N +1, and partition (2N −2n)2

N−2N−n+1(2N )2N−n−1.



In particular, this yields a construction of an infinite family of CURDs with blocksizes 2 and 4, the first such infinite family known:

Corollary 3.12. For all 2N , there exists a CURD with v=2(2N +1), r =2N +1, andpartition 2(2N−1+1)4(2N−2).

By setting l=2, i1=n+m, and i2=m, we get:

Corollary 3.13. For all 2N and m+n<N , there exist a CURD with v=(2N +1)(2n+m−2m), r =2N +1, and partition (2m+n−2m)2

N−2N−m+1(2m+n)2N−n−m(2n−1) and

a CURD with v=2N (2N +1)(2n+m−2m), r =2N +1, and partition (2N −2m+n+2m)2

N−2N−m+1(2N −2m+n)2N−n−m(2n−1)(2N )2

N−m−n−1.

It is interesting to note that the three cases:

• l=1 and i1=1;• l=2, i1=2, and i2=1 and utilizing Theorem 3.1;• l=2, i1=2 and i2=1

can also be constructed by removing one, two, or all but two of the concurrent hyperovalsgiven in Corollary 2.5 from AG(2,2n). Similar constructions on concurrent ovals willbe used later in Theorem 3.19.

C. Subplane Constructions

When considering the {0,1,qb+1}-arcs that are subplanes of PG(2,qa), for b |a∈Z+the kinds of points and their intersection parameters with lines and the arc are shown inTable II. To derive these values, we first notice that in the subplane the number of pointsis q2b+qb+1 and it must have a total of q2b+qb+1 (qb+1)-lines each of whichintersect inside the subplane. Points internal to the subplane cannot be on any 0-linesand, by the definition of a subplane, are on precisely qb+1 (qb+1)-lines. These pointsare on qa+1 lines in the PG(2,qa) and this determines all the parameters for internalpoints. There can be no points external to the arc which are on more than one (qb+1)-line because such lines only intersect inside the subplane. By counting the number ofpoints external to the subplane that appear on (qb+1)-lines, we determine the numbersof the two types of external points. For points that are incident with zero (qb+1)-lines,they are on precisely one tangent for every point of the subplane and this determinesall the parameters for Type 2 points. The single (qb+1)-line through a Type 1 pointforces it to be incident with q2b tangents and thus finishes determining the parametersin Table II. Note that when a=2b, the case of the Baer Subplanes, then there are no0-lines and no external points of the second type.

Theorem 3.14. When q is a prime power, and a=2b then there exists a CURGDD oftype (q2b−qb)q

b+1(q2b)q2b−qb−1 with partition (q2b−qb−1)1(q2b−1)q

2b−1 on q4b−q2b−qb points.

Proof. Remove the arc and a tangent line from PG(2,q2b). The point of tangencydefines the groups. Each other point of the tangent line defines a resolution class. Weobtain the group and block sizes from Table II. �



The following theorem is similar to that given in [18], but we induce a class-uniformlyresolvable structure on it.

Theorem 3.15. If q is a prime power, then there exists a CURGDD of type (q2−q)q+1(q2)1 and partition 11(q+1)q

2−1.

Proof. Choose a point, P , on the Baer subplane, S, in PG(2,q2). The point set ofour CURGDD will be the union of one tangent line, �, through P , (not including thepoint P itself) and the external points of the q+1(q+1)-lines. The lines through Pinduce the groups. The points of any second tangent through P , which must exist sinceqa−qb=q2−q>1, give the resolution classes. Note that when qa−qb>2, there aremultiple resolutions possible. �

When we attempt these two constructions in the case when a>2b, the fact that thereare two different kinds of points off the arc and the fact that all points are incident witha strictly positive number of tangents prevent us from applying the techniques used inTheorem 3.6 to produce strict CURDs; the structures that are produced will have twodifferent kinds of resolution classes.

D. Unital Constructions

A unital is a BIBD(q3+1,q+1,1) and has the parameters b=(q2−q+1)q2 and r =q2.A PG(2,q2) has v′ =b′ =q4+q2+1 lines and points; each point is incident with r ′ =q2+1 lines and each line contains q2+1 points. Thus, when a unital is embedded in thisprojective plane, we get that every point on the unital is on q2(q+1)-lines and exactlyone tangent. This accounts for all lines of the plane and thus the unital is a {1,q+1}-arc.Since every line through a point external to the unital must intersect it, we get only onetype of external point. It is incident with q2−q(q+1)-lines and q+1 tangents.

Theorem 3.16. For all q a prime power, there exists a CURGDD of type q(q2) andpartition 1q(q+1)q

2−q with q2 resolution classes.

Proof. Let V be the point set of a unital embedded in a PG(2,q2) whose point set is V ′.Let P ∈V . The point set of the CURGDD is V \{P}; thus v=q3. The blocks of theunital through P will determine the groups of the CURGDD which give it type q(q2).Let � be the unique tangent line through P . For each P ′ ∈� , P ′ �= P , the lines, otherthan � through P ′, will determine one resolution class of the CURGDD. These pointsare all external to the unital and thus are incident with q+1 tangents. One of these is� and the remaining q others will determine q blocks of size 1 in each class of theCURGDD. The q2−q (q+1)-lines through this external point will determine blocks ofsize q+1 in each resolution class of the CURGDD. There are q2 points on � other thanp and thus q2 resolution classes. �

Theorem 3.17. For all q a prime power, there exists a CURGDD of type (q2−q)(q2)

and partition (q2−q−1)(q2−q)(q2−1)q with q2 resolution classes.

Proof. Let V be the point set of a unital embedded in a PG(2,q2) whose point setis V ′. Let P ∈V and � be the unique tangent to the unital at point P . The point set ofthe CURGDD is V ′′ =V ′ \(V∪�) and thus has size q4+q2+1−(q3+1)−q2=q4−q3.



The lines through P , other than �, determine the q2 groups of size (q2+1)−(q+1)=q2−q . For each P ′ ∈�, P ′ �= P , the lines, other than � through P ′, will determine oneresolution class of the CURGDD. These points are all external to the unital and thus areincident with q+1 tangents. One of these is � and the remaining q others will determineq blocks of size (q2+1)−2=q2−1 in each class of the CURGDD. The q2−q(q+1)-lines through this external point will determine blocks of size (q2+1)−(q+1)−1=q2−q−1 in each resolution class of the CURGDD. There are q2 points on � other thanP and thus q2 resolution classes. �

E. Arcs From RBIBD Constructions

Let q=kt+1. An RBIBD((k−1)q+1,k,1) has b=(kt− t+1)q blocks and r =q reso-lution classes. Since each of the concurrent arcs found by Greig and Rosa described inTheorems 2.8 and 2.9 is a maximal {0,2}-arc on q+1 points, we see that every pointinternal to such an arc is on q 2-lines and zero 0-lines. Since any line external to thearc can only intersect the arc in 2 points or miss it entirely, there is only one set ofintersection parameters for external points. Each of the (k−2)q external points is on(q+1)/2 2-lines and (q−1)/2 0-lines.By removing one arc from those of Theorems 2.8, we get the following.

Theorem 3.18. For every t≥0, there exists a CURD with partition 2(2t+1)4t and aCURD with partition 22t314t .

Proof. Let V be the point set of one of the arcs in an RBIBD(12t+4,4,1) whosepoint set is V ′. The point set of the first CURD will be V ′′ =V ′ \V and thus is of size(12t+4)−(4t+2)=8t+2. The resolution classes of the CURD are simply inheritedfrom the RBIBD. Each line of a resolution class from the RBIBD either hits the arcin 2 points or misses it entirely. All the blocks of the resolution class are disjoint andthey must hit all points in V⊂V ′ and thus we can calculate each resolution class of theCURD that has t blocks of size 4 and 2t+1 blocks of size 2.Let P ∈V be a point of the removed arc. Each block through P was in a different

resolution class in the RBIBD and is therefore in a different resolution class of theCURD. Also each such block must have intersected the arc and thus all have size 2 inthe CURD. Finally, the union of these blocks covers V ′′ ⊂V ′ and thus they satisfy theconditions of Theorem 3.2, giving the second desired CURD. �

When there are more than three concurrent arcs, as in Theorem 2.9, then we canperform some more intricate constructions by removing sets of the concurrent arcs.

Theorem 3.19. If q=kt+1 is a prime power, there exists

1. a CURD with partition (k−2)(q+1)/2k(q−1−2t)/2,

2. a CURD with partition (k−4)tk/4(k−3)1(k−2)tk/2k(tk−4t)/4,

3. a CURD with partition (k−4)tk/4(k−2)tk/2+1k(tk−4t)/4,

4. a CURD with partition 2tk/2314tk/4, and5. a CURD with partition 2(tk+2)/24tk/4,

in the following circumstances:1. For k=4, all t;2. for k=6, t even, t �∈ {2,6}, and t≤832;



3. for k=8, t �∈ {3,11}, and t≤512;4. for k=10, t even, t /∈{4,6,10,12,18,24}, and t≤729.

Proof. Let V be the point set of an RBIBD((k−1)q+1,k,1). Let Vi be the point setsof the k−1 concurrent arcs in the RBIBD((k−1)q+1,k,1), for 1≤ i<k. Let P be thecommon point in the intersection of the arcs.The first CURD is constructed exactly as in Theorem 3.18 by removing a single one

of the concurrent arcs.For the next CURD, we will remove V1 and V2 to get a design on (k−1)q+1−

(q+1)−q=(k−3)q points. It will inherit its r =q resolution classes from the RBIBD.For each block, B, of the RBIBD, define kB1=|B∩V1| and kB2=|B∩V2|. We call theordered pair (kB1,kB2) the type of B and it can take on the values (2,2), (2,0), (0,2)or (0,0). To show that each resolution class in the CURD will have the class-uniformproperty, we show that the resolution classes of the RBIBDs have their blocks uniformlyintersecting the arcs.In Greig’s constructions of the RBIBD((k−1)q+1,k,1)s, they are built from the

associated GDD of type (k−1)q with the groups defined by the blocks through the arcs’concurrent point in the RBIBD [11]. These designs are given as a set of t base blocksto be developed over GF(k−1)×GF(q). These base blocks have the property that thesecond components of the points of the t base blocks exactly cover GF(q)\{0}, andso when developed over GF(k−1) form a holey parallel class for the associated GDD.This GDD is in fact a frame, since developing this class over GF(q) gives a frameresolvable GDD. Additionally, considering only the first component of each element,then the resulting base blocks are duplicates of either

• t copies of the quadratic residues with zero when k≡0 mod 4, or• t/2 copies of the quadratic residues with zero and t/2 copies of the quadraticnon-residues with zero,

with some such blocks appearing multiple times. Ignoring the duplicates of any suchblock, and still considering only the first component, developing these base blocks givesa BIBD(k−1,k/2,kt/4). Since these k−1 first components identify the arcs [13], wesee that each holey parallel class contains kt/4 lines through both of the first two arcs,that is blocks of type (2,2), and so kt/4 lines through the first arc but not the second,blocks of type (2,0) [11, 13].One block from the resolution class, BP , must contain the point P and thus is type

(2,2). In each intersection, P is one of the two points; thus, this block intersectsV1∪V2 in precisely three points and thus gives a block of size k−3 in the CURD.Since V1 and V2 are disjoint except for P , each of the other kt/4 blocks of type(2,2) must intersect V1∪V2 in four points and contribute blocks of size k−4 to theCURD. Let na =kt/4 denote the number of blocks in a resolution class, other thanBP of type (2,2) and nb=kt/4, nc, nd the number of types (2,0), (0,2) and (0,0),respectively. Using the fact that these together form a resolution class excluding BP ,we get

na+nb+nc+nd = kt− t (4)

na+nb = kt/2 (5)

na+nc = kt/2 (6)



yielding

na = kt/4 (7)

nb=nc = kt/4 (8)

nd = t (k−4)/4 (9)

as desired.In this CURD, we can see that all blocks which contained P are all of the same

size, each is in a separate resolution class of the CURD and together form a partitionof the point set of the CURD and thus by Theorem 3.2, we get a CURD of partition(k−4)tk/4(k−2)tk/2+1k(tk−4t)/4.The fourth and fifth desired CURDs are simply the complement of the previous

two in the RBIBD. They can equivalently be constructed by removing all but twoarcs from the RBIBD, then adding back in P for the fourth and removing all buttwo arcs from the RBIBD for the fifth. The fifth can also be derived from the fourthby applying Theorem 3.1. �

It is possible that Theorem 3.19 can be applied more generally: delete sets of morethan three of the concurrent arcs. The partitions of the CURDs depend on which setsof arcs get deleted and we leave a fully general statement of this result as an openproblem.

ACKNOWLEDGMENT

The first and third authors would like to thank NSERC for financial research support.

REFERENCES

[1] B. Alspach, H. Gavlas, M. Sajna, and H. Verrall, Cycle decompositions. IV. Complete directedgraphs and fixed length directed cycles, J Combin Theory Ser A 103(1) (2003), 165–208.

[2] T. Beth, D. Jungnickel, and H. Lenz, Design Theory, 1, 2nd edn, Cambridge University Press,Cambridge, 1999.

[3] D. Bryant and P. Danziger, On bipartite 2-factorsations of Kn− I and the Oberwolfachproblem, preprint.

[4] C. J. Colbourn and J. H. Dinitz (Editors), Handbook of Combinatorial Designs, 2nd edn,Discrete Mathematics and Its Applications (Boca Raton), Chapman & Hall/CRC, Boca RatonFL, 2007.

[5] P. Danziger and B. Stevens, Class-uniformly resolvable designs, JCD 9 (2001), 79–99.

[6] P. Danziger and B. Stevens, Class-uniformly resolvable group divisible structures. I. Resolvablegroup divisible designs, Electron J Combin 11(1) (2004), 13, Research Paper 23 (electronic).

[7] P. Danziger and B. Stevens, Class-uniformly resolvable group divisible structures. II. Frames,Electron J Combin 11(1) (2004), 17, Research Paper 24 (electronic).

[8] P. Dembowski, Finite Geometries, Springer, Berlin, 1997, reprint of the 1968 original.

[9] R. H. F. Denniston, Some maximal arcs in finite projective planes, J Combin Theory 6 (1969),317–319.



[10] J. C. Fisher, J. W. P. Hirschfeld, and J. A. Thas, Complete arcs in planes of square order,Ann Discrete Math 30 (1984), 243–250.

[11] M. Greig, Some group divisible design constructions, JCMCC 27 (1998), 33–52.

[12] M. Greig, Designs from projective planes and PBD bases, JCD 7 (1999), 341–373.

[13] M. Greig and A. Rosa, Maximal arcs in Steiner systems S(2,4,v), Discrete Math 267(1–3)(2003), 143–151, Combinatorics 2000 (Gaeta).

[14] P. Gvozdjak, On the Oberwolfach problem for cycles with multiple lengths, PhD thesis, SimonFraser University, Burnaby, 2003.

[15] M. S. Keranen, R. S. Rees, and A. C. H. Ling, An infinite class of fibres in CURDs withblock sizes two and three, J Combin Des 12(1) (2004), 46–71.

[16] E. Lamken, R. Rees, and S. Vanstone, Class-uniformly resolvable pairwise balanced designswith block sizes two and three, DM 92 (1991), 197–209.

[17] M. Sajna, Cycle decompositions. III. Complete graphs and fixed length cycles, J Combin Des10(1) (2002), 27–78.

[18] W. D. Wallis, Configurations arising from maximal arcs, J Combin Theory Ser A 15 (1973),115–119.

[19] C. Wang, Y. Tang, and P. Danziger, Resolvable modified group divisible designs with blocksize three, J Combin Des 15(1) (2007), 2–14.

[20] D. Wevrick and S. A. Vanstone, Class-uniformly resolvable designs with block sizes 2 and 3,JCD 4 (1996), 177–202.


Documents

Geometrical constructions of class-uniformly resolvable structure