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Cyclic and Bicyclic Decompositions of the Complete Graph into the 4-Cycle with a Pendant Edge

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*. *Presenter, East Tennessee State University, Department of Mathematics and Statistics. 2010 Southeastern MAA Conference

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Cyclic and Bicyclic Decompositions of the Complete Graph into the 4-Cycle with a Pendant Edge

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  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* *Presenter, East Tennessee State University, Department of Mathematics and Statistics 2010 Southeastern MAA Conference Elon University; Elon, NC March 26, 2010

  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 { G1, G2, … , Gn} where GiG and V(Gi) V(H) for all i, E(Gi) ∩ E(Gj) = Ø if i ≠ j, and Gi = H.

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

  5. 1 2 0 (0,1,3) (1,2,4) 3 1 Example. There is a decomposition of K7 into 3-cycles: 6 (2,3,5) (3,4,6) (4,5,0) (5,6,1) 2 (6,0,2) 5 3 4

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

  7. Jakob Steiner 1796-1863 v ≡ 1 or 3 (mod 6) is necessary. From the Saint Andrews MacTutor History of Mathematics website. J. Steiner, Combinatorische Aufgabe, Journal für die Reine und angewandte Mathematik (Crelle’s Journal), 45 (1853), 181-182.

  8. Theorem.A STS(v) exists if and only if v ≡ 1 or 3 (mod 6). Note. Sufficiency follows from Reiss. 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.

  9. Thomas P. Kirkman 1806-1895 STS(v) iff v ≡ 1 or 3 (mod 6). 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.

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

  11. Theorem.An L-decomposition of Kv exists if and only if v ≡ 0 or 1 (mod 8). Jean-Claude Bermond From Bermond’s website: http://www-sop.inria.fr/members/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.

  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. Theorem.An H-decomposition of Kv exists if and only if v ≡ 0 or 1 (mod 5) and v≥ 11. Dominique Sotteau Alex Rosa From: http://www.d.umn.edu/~dfroncek/alex/ and http://www-direction.inria.fr/international/DS/page_personnelle.html J. C. Bermond, C. Huang, A. Rosa, and D. Sotteau, Decompositions of Complete Graphs into Isomorphic Subgraphs with Five Vertices, Ars Combinatoria10 (1980), 211-254.

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

  15. 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.

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

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

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

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

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

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

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

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

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

  27. Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles. 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 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 Mathematica54 (1998), 51-57.

  31. 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-matica54 (1998), 51-57.

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

  33. 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.

  34. A Cyclic H-Decomposition of K11 0 5 4 1 10 (5, 3, 0, 1) - 10 2 31 9 2 3 8 4 7 6 5

  35. Theorem. A bicyclic H-decomposition of Kv, exists if and only if (i) M = N ≡ 3 (mod 10), M = N ≥ 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 K26 With M = N = 13.

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

  38. Thanks, eh! Good Day, eh!

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