1. Balanced Path Decompositions of Crowns and Directed Crowns Hung-Chih Lee and Shun-Li Hsu Department of Information Technology Ling Tung University Taichung, Taiwan 40852, R.O.C.

2. Outline Introduction Previous results Our results

3. Introduction H-decomposition of G Balanced H-decomposition of G

4. H-decomposition of G Let G and H be graphs (digraphs). An H- decomposition of G is a partition of the edge (arc) set of G into subsets each of which induces a graph (digraph) isomorphic to H. If G has an H-decomposition, then we say H decomposes G, denoted by H|G.

5. Balanced H-decomposition of G An H-decomposition of G is balanced if each vertex of G belongs to the same number of members in the decomposition. We write H||G If G admits a balanced H- decomposition.

6. Path and directed path : a path on k vertices : a directed path on k vertices P4P4

7. Crown :

8. Properties of the crown

9. Directed crown

10. Previous results Bermond （ 1975 ） Bermond Hung and Mendelsohn （ 1977 ） Hung and Mendelsohn Lee and Lin （ 2009 ） Lee and Lin

11. Previous results Bermond （ 1975 ）

12. Previous results Hung and Mendelsohn （ 1977 ）

13. Previous results Lee and Lin （ 2009 ）

14. Main results: Theorem A. Corollary B.

15. Main results: Theorem C. Corollary D.

16. label of edges (arcs)

17. Example Label 0 :Label 1 :Label 2 :

18. Example Label 0 : Label 1 :

19. Notations Eaxmple

20. Notations Example

21. Notations

22. Lemmas Lemma 2.1

23. Lemmas Lemma 2.2

24. Lemmas Lemma 2.3

25. Lemmas Lemma 2.4 Example

26. Lemmas Lemma 2.5

27. Proof of Theorem A (Necessity) C n,l is l-regular and by Lemma 2.5Lemma 2.5 (Sufficiency) 2(k － 1)|lk ⇒ k － 1|l Case 1. k is even By Lemma 2.2, it suffices to show

28. Base graph 0 1 2

29. Proof of Theorem A Case 2. k is odd 2(k － 1)|lk ⇒ 2(k － 1)|l. By Lemma 2.2, it suffices to show

30. Proof of Theorem A

31. 02 1 3 Baes graph

32. Proof of Theorem B (Necessity) C * n,l is 2l-regular and by Lemma 2.5 (Sufficiency) Case 1. 2(k － 1)|lk By Theorem A, there exists a balanced P k - decomposition  of C * n,l Replace each edge in C n,l by two arcs with opposite directions ⇒ each P k in  becomes two with opposite directions ⇒ Done.

33. Base graph 0 2 Replace each edge in the crown by two arcs with opposite directions

34. Proof of Theorem B Case 2.

35. Proof of Theorem B

36. Base graph

37. Thank you for your attention!

38. Previous results Path decomposition Tarsi （ 1983 ） Tarsi Truszczyński （ 1985 ） Truszczyński Shyu and Lin （ 2003 ） Shyu and Lin Meszka and Skupień （ 2006 ） Meszka and Skupień Balanced path decomposition Bermond （ 1975 ） Bermond Hung and Mendelsohn （ 1977 ） Hung and Mendelsohn

39. Path decomposition Tarsi （ 1983 ）

40. Path decomposition Truszczyński （ 1985 ）

41. Path decomposition Shyu and Lin （ 2003 ）

42. Path decomposition Meszka and Skupień （ 2006 ）

43. Our object Find the necessary and sufficient conditions for

44. Main results Necessary condition － Counting Method Necessary condition Sufficient condition － Construction Method

45. Procedure of the proof Labeling the edges of the crown Find a base graph (path) k is even k is odd 2(k-1) does not divide lk Shifting the base graph

46. Future works Find the necessary and sufficient conditions for

47. , 24 C G

48. 9 prisoners problem (P 3 ||K 9 )