1. 1 黃國卿 靜宜大學應用數學系

2. 2 Let G be an undirected simple graph and H be a subgraph of G. G is H-decomposable, denoted by H|G, if its edge set E(G) can be decomposed into subgraphs such that each subgraph is isomorphic to H. G has an H-decomposition if G is H- decomposable. Definitions

3. 3

4. 4 butis not -decomposable for other of size 3.

5. 5 Motivation (Chartrand, Saba and Mynhardt;1994) If is a 2-connected graph of order sizethenis-decomposable. and Conjecture 1. If is a graph of size and then is decomposable for some graph - decomposable for some graph size 3. of (Chartrand, Saba and Mynhardt;1994) Conjecture 2.

6. 6 It is interesting to us for studying the H- decompositions of a graph G with H of size at most three.

7. 7 It is trivial that for any graph with at least one edge.

8. 8 or (Chartrand, Polimeni and Stewart.) Every nontrivial connected graph of even size is -decomposable. Theorem 1. (Chen and Huang) Theorem 2. Suppose G is a graph of even size and different from K 3 ∪ K 2. Then G is M 2 -decomposable if and only if q(G) ≧ 2Δ(G).

9. 9 The Conjecture 1 is not true in general.orCounterexamples (C. Sunil Kumar)

10. 10 except or except or Theorem 3. (C. Sunil Kumar)

11. 11 -packings of graphs. 1. -packings of graphs. Main results -decompositions of graphs with H of size three. 2. H-decompositions of graphs with H of size three. 3. -decomposability of graphs.

12. 12-packing. for : leave If then are mutually disjoint.

13. 13-packing

14. 14 Suppose is a graph different from with and Then has a -packing with leave L where and Then has a -packing with leave L where Theorem 4. -packings of graphs. 1. -packings of graphs.

15. 15 (1) and (2-regular) (2) and (3) and (3-regular) (4) and Proof. Induction on

16. 16 If is a graph of size and then is decomposable for some graph - decomposable for some graph size 3. of Conjecture 2.

17. 17 Theorem 5. The Conjecture 2 is affirmative. Proof. (1) If q(G) = 3, then G|G. (2) If G = K 4 or K 1,1,3c+1, then P 4 |G. (3) Otherwise, by Theorem 4, we have (P 3 ∪ P 2 )|G.

18. 18 H-decompositions of complete multipartite graphs. 2.1 H-decompositions of complete multipartite graphs. H-decompositions of cubic graphs. 2.2 H-decompositions of cubic graphs. H-decompositions of hypercubes. 2.3 H-decompositions of hypercubes. -decompositions of graphs with H of size three. 2. H-decompositions of graphs with H of size three.

19. 19 2.1 H-decompositions of complete multipartite graphs. andor 1 andor 2 and or 3

20. 20 The K 3 -decomposability of complete multipartite graphs is still widely open.

21. 21 2.2 H-decompositions of cubic graphs. A cubic graph is a 3-regular graph. Let be a cubic graph. By the degree-sum formula, we obtain Hence,

22. 22Suppose is a cubic graph. is a cubic graph. (1) is not -decomposable. (2) is -decomposable if is 2-connected. (3) is -decomposable if and only if it is bipartite. (5) is -decomposable except (4) is -decomposable except Theorem 6.

23. 23If is a 2-connected graph of order sizethenis-decomposable.and Conjecture 1. (Chen and Huang) If is 2-connected, and size then is -decomposable. Conjecture 3.

24. 24 2.3 H-decompositions of hypercubes. (1) An -cube, denoted by and for and for (2) (3) is bipartite is not -decomposable., is defined recursively by

25. 25 (4) for (5)

26. 26

27. 27 Suppose and is a graph of size 3. Then is -decomposable if is different from Theorem 7.

28. 28 3. -decomposability of graphs. (1) If then (2) If then is not -decomposable. (3) Suppose Then Letand Theorem 8.

29. 29 Theorem 9. For a simple graph Theorem 10. Suppose is a simple graph and Then is equitably -edge colorable. (Vizing) (De Werra)

30. 30 (1) If then (2) If then is not -decomposable. (3) Suppose Then Letand Corollary 11.

31. 31 W.l.o.g., assume (1) (2) or or Proof of Proof of.3

32. 32 Conjecture 4. If and then

33. 33 Theorem 12. Theorem 13. Suppose G is a graph of size q(G) = 2Δ(G). Then χ’(G) = Δ(G) + 1 if and only if G = K 3 ∪ K 2. Suppose G is a graph of size q(G) = 3Δ(G) and Δ(G ) ≧ 5. Then χ’(G) = Δ(G).

34. 34 Remark For there is a graph such that and and

35. 35 Proof.Let Case1. Then where Case2. Then where is maximum where is maximum matching of and matching of and

36. 36 ExampleCase1. Case2. For

37. 37

38. 38 結論. Conjecture 00. (C.Sunil Kumar) If is 3-connected and size then is -decomposable. Theorem 00. If is a 2-connected cubic graph, then is -decomposable.

39. 39 Conjecture 00. (Chen, Huang and Tsai) If is 2-connectedand, and size then is -decomposable. then is -decomposable. Remark 00. Coniecture4 Coniecture3

40. 40 (3) is 3-regular. (3) is 3-regular. (a) connected. (a) connected. (b) disconnected. (b) disconnected.Proof.

41. 41No!! Basic step : (a) connected.

42. 42Let Induction step :

43. 43

44. 44 (b) disconnected where and Let

45. 45

46. 46 2 or 3