Student: Ying-Yen Hsu Advisor: Hung-Lin Fu Department of Applied Mathematics

1. Student: Ying-Yen Hsu Advisor: Hung-Lin Fu Department of Applied Mathematics National Chiao Tung University

2. Definitions The maximal path-packings of complete graph

3. Definitions • An H-packing of a graph Gis a set such that is isomorphic to H of G for where and are edge-disjoint for all . • The leaveL of a packing is the subgraph induced by the set of edges of G that does not occur in any . • If , then G is said to be H-decomposable, i.e., H | G. The maximal path-packings of complete graph

4. Definitions b0 a0 a1 b1 a5 b5 a2 b2 a4 b4 a3 b3 P6-packing of K6 p1=a5a0a1a3a4a2, p2=a2a1a5a4a0a3, ｛p1, p2｝is a P6-packing of K6. p3=b0b1b3b5b2b4, p4=b2b1b5b4b0b3, p5=b1b4b3b2b0b5, ｛p3, p4, p5｝is a P6-decomposition of K6. The maximal path-packings of complete graph

5. Definitions c0 a0 b0 a1 b1 c1 a5 b5 c5 a2 b2 c2 a4 b4 c4 c3 a3 b3 The maximal path-packings of complete graph A packing is said to be maximal if the leave of contains no subgraph that is isomorphic to H. Consider maximal Pk+1-packing of Kn.

6. Definitions The size of a packing , denoted by , is the cardinality of . is a maximum maximal packing (or simply maximum packing) means that for all other maximal packing . is a minimum maximal packing (or simply minimum packing) means that for all other maximal packing . The maximal path-packings of complete graph

7. Definitions • The spectrumS(G;H) denoted the set of all sizes such that there exists a maximal H-packing of G with this size. • maxS(G;H)is the size of the maximum packing. • minS(G;H)is the size of the minimum packing. • We use S(n; H) to denote S(Kn;H) for convenience. The maximal path-packings of complete graph

8. Maximum (maximal) packing The maximal path-packings of complete graph

9. Maximum (maximal) packing There exists a maximum (maximal) Pk+1-packing of Kn. Moreover, the size of the packing is . That is, . The maximal path-packings of complete graph

10. Minimum (maximal) packing The maximal path-packings of complete graph

11. Lower bound A minimum packing has a maximum number of edges of leave. is a maximal if and only if L contains no . Hence, we consider the maximum number of edges of leave which contains no for the minimum problem. The maximal path-packings of complete graph

12. Lower bound k k r t Definition. For a given graph F, ext(n; F)denote the maximum number of edges of a graph of order n not containing F as a subgraph. Consider n = tk+ r, 0 ≤ r < k, then . Moreover, the graph with the edge number is [2]. The maximal path-packings of complete graph

13. Lower bound Corollary 1. Consider n = tk+ r, 0 ≤ r < k. If is a minimum (maximal) of and L is a leave of , then . I.e. . The maximal path-packings of complete graph

14. Upper bound Theorem 2. Considerk ≤ 6, n = tk + r ≥ k + 1, 0 ≤ r < k, there exist a minimum (maximal) of where The maximal path-packings of complete graph

15. Upper bound Kn k k r k • The idea of the proof of Theorem 2. • Keep the leave L is the graph and consider the subgraphs or . • By induction on n The maximal path-packings of complete graph

16. Upper bound Consider n = tk + r ≥ k + 1. If r = 0, ? If , ? If , ? The maximal path-packings of complete graph

17. Upper bound x0 x1 x2 x3 x4 x0 x1 x2 xk-1 xk-1 … 4 1 0 2 3 … y0 y1 y2 y3 y4 Y0 y1 y2 yk-1 yk-1 diff (xi , yj) = j－i(mod 5) Consider There exists a -decomposition of . The maximal path-packings of complete graph

18. Upper bound x0 x1 x2 x3 x4 x0 x1 x2 x3 x4 z0 y0 y1 y2 y3 y4 y0 y1 y2 y3 y4 Consider , The maximal path-packings of complete graph

19. Upper bound x0 x1 x2 x3 x4 z0 z1 y0 y1 y2 y3 y4 There exists a -decomposition of where . The maximal path-packings of complete graph

20. Upper bound Consider , and both k and r are odd The maximal path-packings of complete graph

21. Upper bound z0 z1 z2 x0 x1 x2 x3 x4 y0 y1 y2 y3 y4 x0 x1 x2 x3 x4 z0 z1 z2 y0 y1 y2 y3 y4 The maximal path-packings of complete graph

22. Upper bound If k ≤ 6, there exists a -decomposition of where and both of k and r are odd. Open problem: Is it true when k > 6? The maximal path-packings of complete graph

23. Upper bound x0 x1 x2 x3 x0 x1 x2 x3 o1 o2 o y0 y1 y2 y3 y0 y1 y2 y3 Consider , and one of k and r is even The maximal path-packings of complete graph

24. Upper bound If k ≤ 6, there exists a -decomposition of where and one of k and r is even. Open problem: Is it true when k > 6? Remark. In Truszczynski's paper [9], he verified that has -decomposition if and both of k and r are even. He also have the proof when , k is odd and r is even. The maximal path-packings of complete graph

25. Upper bound k k r k If , ? The maximal path-packings of complete graph

26. Upper bound x0 x1 x2 x3 x4 x0 x1 x2 x3 x4 y0 y1 y2 y3 y4 y5 y0 y1 y2 y3 y4 y5 y6 Consider , The maximal path-packings of complete graph

27. Upper bound If k ≤ 6, there exists a -decomposition of where and one of k and r is even. Open problem: Is it true when k > 6? The maximal path-packings of complete graph

28. Upper bound n: otherwise The maximal path-packings of complete graph

29. Upper bound 0 0 0 6 1 7 1 1 5 5 2 6 2 2 4 4 3 5 3 3 4 The maximal path-packings of complete graph If k ≤ 6, n=k+r and either or k,r odd, there exists a of where .

30. Maximum (maximal) packing Theorem 2. Considerk ≤ 6, n = tk + r ≥ k + 1, 0 ≤ r < k, there exist a minimum (maximal) of where The maximal path-packings of complete graph

31. Spectrum of the packing The maximal path-packings of complete graph

32. Spectrum of the packing Theorem 3. Suppose n = tk + r ≥ k + 1, 0 ≤ r < k. If k ≤ 6, then The maximal path-packings of complete graph

33. Spectrum of the packing k k k k r Kn k k k k Kk, …, k, r If r = 0, consider Kk,k. Otherwise, consider Kk,r and Kk,k. The maximal path-packings of complete graph

34. Spectrum of the packing x0 x1 x2 x3 x4 x0 x1 x2 x3 x4 y0 y1 y2 y3 y4 y0 y1 y2 y3 y4 x0 x1 x2 x3 x4 x0 x1 x2 x3 x4 y0 y1 y2 y3 y4 y0 y1 y2 y3 y4 The maximal path-packings of complete graph

35. Spectrum of the packing x0 x1 x2 x3 x4 x0 x1 x2 x3 x4 y0 y1 y2 y3 y4 y0 y1 y2 y3 y4 x0 x1 x2 x3 x4 x0 x1 x2 x3 x4 y0 y1 y2 y3 y4 y0 y1 y2 y3 y4 The maximal path-packings of complete graph

36. Spectrum of the packing Theorem 3. Suppose n = tk + r ≥ k + 1, 0 ≤ r < k. If k ≤ 6, then The maximal path-packings of complete graph

37. References The maximal path-packings of complete graph

38. References [1] A.E. Brouwer, Optimal packing of K4's into a Kn, J. Combin. Theory Ser. A 26 (1979) 278 - 297. [2] BëlaBollobäs, Graduate Texts in Mathematics: Modern Graph Theory, Published by Springer (1998). [3] D. Bryant, Packing paths in complete graphs, J. Combin. Theory Ser. B 100 (2010) 206-215. [4] Douglas B.West, Introduction to Graph Theory, Published by Prentice Hall (2001). [5] C.C. Chen, H.L. Fu and K.C Huang, (P3∪P2)-packing of graphs. The maximal path-packings of complete graph

39. References [6] H.L. Fu, K.C. Huang and C.L Shiue, Maximal congurations of stars, JCMCC 8 (1990) 51 - 59. [7] Y. Roditty, Packing and covering of the complete graph with a graph G of four vertices or less, J. Combin. Theory Ser. A 34 (1983) 231 - 243. [8] Y. Roditty, Packing and covering of the complete graph IV, the trees of order 7, ArsCombinatoria 35 (1993) 33 - 64. [9] A. Rosa and S.Znam, Packing pentagons into complete graphs: how clumsy can you get, Discrete Math. 128 (1994), 305 - 316. The maximal path-packings of complete graph

40. References [10] M. Tarsi, Decomposition of a complete multigraph into simple paths: Nonbalancedhandcued designs, J. Combin. Theory Ser. A 34 (1983) 60 - 70. [11] M. Truszczynski, Note on the decomposition of λKm,n ( ) into paths, Discrete Math. 55 (1985), 89-96. [12] J. Schonheim, On maximal systems of k-tuples, Studia Sci. Math. Hunger. (1996), 363 - 368. [13] J. Schonheim and A. Bialostocki, Packing and covering of the complete graph with 4-cycles, Canadian Math. Bull. (1975), 703 – 708. The maximal path-packings of complete graph

41. Thank you for your attention! The maximal path-packings of complete graph