Cyclic and Bicyclic Decompositions of the Complete Graph into the 4-Cycle

with a Pendant Edge

Daniel “Lupo” Cantrell Gary “Hoser” CokerRobert “Knob” Gardner*

2010 Southeastern MAA ConferenceElon University; Elon, NC

March 26, 2010

*Presenter, East Tennessee State University, Department of Mathematics and Statistics

Act 1. Decompositions

Steiner Triple Systems

Jakob Steiner

1850s

Definition. A decomposition of a simple graph H with isomorphic copies of graph G is a set

{ G1, G2, … , Gn}

where Gi G and V(Gi) V(H) for all i, E(Gi) ∩ E(Gj) = Ø if i ≠ j, and

Gi = H.

n

i 1

Example. There is a decomposition of K5 into 5-cycles.

= U

Example. There is a decomposition of K7 into 3-cycles: 1 2

5 2

0

16

34

(0,1,3)

(1,2,4)

(2,3,5)(3,4,6)(4,5,0)(5,6,1)(6,0,2)

3

Definition. A Steiner triple system of order v, STS(v), is a decomposition of the complete graph on v vertices, Kv , into 3-cycles.

Note. We shall restrict today’s presentation to decompositions of complete graphs.

From the Saint Andrews MacTutor History of Mathematics website.

Jakob Steiner

1796-1863

J. Steiner, Combinatorische Aufgabe, Journal für die Reine und angewandte Mathematik (Crelle’s Journal), 45 (1853), 181-182.

v ≡ 1 or 3 (mod 6) is necessary.

M. Reiss, Über eine Steinersche combinatorsche Aufgabe welche in 45sten Bande dieses Journals, Seite 181, gestellt worden ist, Journal für die Reine und angewandte Mathematik (Crelle’s Journal), 56 (1859), 326-344.

Theorem. A STS(v) exists if and only if v ≡ 1 or 3 (mod 6).

Note. Sufficiency follows from Reiss.

Thomas P. Kirkman

1806-1895

From the Saint Andrews MacTutor History of Mathematics website.

T. Kirkman, On a problem in combinations, Cambridge and Dublin Mathematics Journal, 2 (1847), 191-204.

STS(v) iff v ≡ 1 or 3 (mod 6).

= L

Definition. The 3-cycle with a pendant edge is denoted L and is:

The graph L is sometimes called the lollipop.

From Bermond’s website: http://www-sop.inria.fr/members/Jean-Claude.Bermond/

Jean-Claude Bermond

J. C. Bermond and J. Schonheim, G-Decompositions of Kn where G has Four Vertices or Less, Discrete Math. 19 (1977), 113-120.

Theorem. An L-decomposition of Kv exists if and only if v ≡ 0 or 1 (mod 8).

Definition. The 4-cycle with a pendant edge is denoted H and is:

= H

The graph H is sometimes called a kite. We call H, for personal reasons, the Hoser graph.

From: http://www.d.umn.edu/~dfroncek/alex/ and http://www-direction.inria.fr/international/DS/page_personnelle.html

Alex Rosa

J. C. Bermond, C. Huang, A. Rosa, and D. Sotteau, Decompositions of Complete Graphs into Isomorphic Subgraphs with Five Vertices, Ars Combinatoria 10 (1980), 211-254.

Theorem. An H-decomposition of Kv exists if and only if v ≡ 0 or 1 (mod 5) and v ≥ 11.

Dominique Sotteau

Act 2. Automorphisms

Cycles and Bicycles

Peltesohn and Gardner

1930s to present

Automorphisms, eh!

Take off!

Definition. An automorphism of a G-decomposition of H is a permutation of V(H) which fixes the set of copies of G, { G1, G2, … , Gn}.

Recall. A permutation can be classified by its disjoint decomposition into cycles.

Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.

Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.

Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.

Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.

Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.

Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.

Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.

MN

Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.

MN

Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.

MN

Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.

MN

Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.

MN

Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.

MN

Theorem. A STS(v) admitting a cyclic automorphism exists if and only if

v ≡ 1 or 3 (mod 6), v ≠ 9.

R. Peltesohn, A Solution to Both of Heffter's Difference Problems (in German), Compositio Math. 6 (1939), 251-257.

Theorem. A bicyclic Steiner Triple System of order v exists if and only if v = M + N ≡ 1 or 3 (mod 6), M ≡ 1 or 3 (mod 6), M ≠ 9 (M > 1), and M | N.

R. Calahan and R. Gardner, Bicyclic Steiner Triple Systems, Discrete Math. 128 (1994), 35-44.

Theorem. A cyclic L-decomposition of Kv exists if and only if v ≡ 1 (mod 8).

J. C. Bermond and J. Schonheim, G-Decompositions of Kn where G has Four Vertices or Less, Discrete Math. 19 (1977), 113-120.

R. Gardner, Bicyclic Decompositions of Kv into Copies of K3 {e}, Utilitas Mathematica 54 (1998), 51-57.

Theorem. A bicyclic L-decomposition of Kv exists if and only if (i) N = 2 M and v = M + N ≡ 9 (mod 24), or (ii) M ≡ 1 (mod 8) and N = k M where k ≡ 7 (mod 8).

R. Gardner, Bicyclic Decompositions of Kv into Copies of K3 {e}, Utilitas Mathe-matica 54 (1998), 51-57.

Act 3. New Results

Hoser Graphs

Cantrell, Coker, Gardner

2010

Theorem. A cyclic H-decomposition of Kv exists if and only if v ≡ 1 (mod 10).

D. Cantrell, G. D. Coker, and R. Gardner, Cyclic, f-Cyclic, and Bicyclic Decompositions of the Complete Graph into the 4-Cycle with a Pendant Edge, Utilitas Mathematica,to appear.

A Cyclic H-Decomposition of K11

01

2

3

4

567

8

9

10 (5, 3, 0, 1) - 102 3 1

54

Theorem. A bicyclic H-decomposition of Kv, exists if and only if (i) M = N ≡ 3 (mod 10), =≥ 13, or(ii) M ≡ 1 (mod 10) and N = k M where k ≡ 9 (mod 10).

D. Cantrell, G. D. Coker, and R. Gardner, Cyclic, f-Cyclic, and Bicyclic Decompositions of the Complete Graph into the 4-Cycle with a Pendant Edge, Utilitas Mathematica,to appear.

A Bicyclic H-decomposition of K26 With M = N = 13.

Special Thanks To: Elsinore Beer for the inspiration for this research!

Good Day, eh!