1. Cyclic and Bicyclic Decompositions of the Complete Graph into the 4-Cycle with a Pendant Edge Daniel “Lupo” Cantrell Gary “Hoser” Coker Robert “Knob” Gardner* 2010 Southeastern MAA Conference Elon University; Elon, NC March 26, 2010 *Presenter, East Tennessee State University, Department of Mathematics and Statistics

2. Act 1. Decompositions Steiner Triple Systems Jakob Steiner 1850s

3. Definition. A decomposition of a simple graph H with isomorphic copies of graph G is a set { G 1, G 2, …, G n } where G i G and V(G i ) V(H) for all i, E(G i ) ∩ E(G j ) = Ø if i ≠ j, and G i = H.

4. Example. There is a decomposition of K 5 into 5-cycles. = U

5. Example. There is a decomposition of K 7 into 3-cycles: 12 5 2 0 1 6 3 4 (0,1,3) (1,2,4) (2,3,5) (3,4,6) (4,5,0) (5,6,1) (6,0,2) 3

6. Definition. A Steiner triple system of order v, STS(v), is a decomposition of the complete graph on v vertices, K v, into 3-cycles. Note. We shall restrict today’s presentation to decompositions of complete graphs.

7. 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.

8. 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.

9. 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).

10. = L Definition. The 3-cycle with a pendant edge is denoted L and is: The graph L is sometimes called the lollipop.

11. 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 K n where G has Four Vertices or Less, Discrete Math. 19 (1977), 113-120. Theorem. An L-decomposition of K v exists if and only if v ≡ 0 or 1 (mod 8).

12. 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.

13. 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 K v exists if and only if v ≡ 0 or 1 (mod 5) and v ≥ 11. Dominique Sotteau

14. Act 2. Automorphisms Cycles and Bicycles Peltesohn and Gardner 1930s to present Automorphisms, eh! Take off!

15. Definition. An automorphism of a G- decomposition of H is a permutation of V(H) which fixes the set of copies of G, { G 1, G 2, …, G n }. Recall. A permutation can be classified by its disjoint decomposition into cycles.

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

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

23. M N

24. M N

25. M N

26. M N

27. M N

28. 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.

29. 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.

30. Theorem. A cyclic L-decomposition of K v exists if and only if v ≡ 1 (mod 8). J. C. Bermond and J. Schonheim, G-Decompositions of K n where G has Four Vertices or Less, Discrete Math. 19 (1977), 113-120. R. Gardner, Bicyclic Decompositions of K v into Copies of K 3 {e}, Utilitas Mathematica 54 (1998), 51-57.

31. Theorem. A bicyclic L-decomposition of K v 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 K v into Copies of K 3 {e}, Utilitas Mathe- matica 54 (1998), 51-57.

32. Act 3. New Results Hoser Graphs Cantrell, Coker, Gardner 2010

33. Theorem. A cyclic H-decomposition of K v 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.

34. A Cyclic H-Decomposition of K 11 0 1 2 3 4 56 7 8 9 10 (5, 3, 0, 1) - 10 2 3 1 5 4

35. Theorem. A bicyclic H-decomposition of K v, 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.

36. A Bicyclic H-decomposition of K 26 With M = N = 13.

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

38. Good Day, eh!