Packing Graphs with 4-Cycles

1. Packing Graphs with 4-Cycles 學生: 徐育鋒 指導教授: 高金美教授 2013組合新苗研討會 (2013.08.10 ~ 2013.08.11) 國立高雄師範大學

2. 1. Definition 2. Known Results 3. 4-Regular Graphs 4. Main Results 5. Future Works

3. Definition

4. Definitions 1. A graph G is an order pair (V, E), where V is a non-empty set called a vertex set and E is a set of two-element subsets of V called an edge set. 2. degG(v) = the number of edges incident with a vertex v in G. 3. If all the vertices of a graph have the same degree r, then the graph is called r-regular.

5. Definitions v6 v1 V = {v1, v2, v3, v4, v5, v6}. E = {v1v2, v1v3, v1v5, v1v6, v2v3, v2v4, v2v6, v3v4, v3v5, v4v5, v4v6, v5v6}. G: v5 v2 v4 v3 The graph G is 4-regular.

6. Definitions 5. Cn = (v1,v2, ..., vn) : n-cycle v1 v5 v2 v4 v3 C5=(v1, v2, v3, v4, v5)

7. v1 Definitions v5 v2 6. Kn : the complete graph of order n. v4 v3 K5

8. Definitions 7. KU,V: the complete bipartite graph with partite set U, V. If|U| = m, |V| = n, then KU,V can be denoted byKm,n. v1 v2 v3 v4 v5 v6 U = {v1, v2, v3}, V = {v4, v5, v6} KU,V = K3,3

9. Definitions Let = {H1, H2, , Hs} be a set of subgraphs of G. If E(H1)  E(H2)    E(Hs) = E(G) and E(Hi)  E(Hj) =  for i j, then we call  is a decomposition (packing) of G. If Hi is isomorphic to a subgraphH of G for each i= 1, 2, , s, then we say that G has an H decomposition (H system) or  is a H packing of G. If Hi is isomorphic to a subgraphH of G for each i= 1, 2, , s–1, then we say that G can be packed with H and leave Hs. That is, G – E(Hs)has an H decomposition. and leave Hs. That is,  – {Hs} is a H packing of G – E(Hs).

10. v1 v1 v2 v2 v3 v3 H1: G: v4 v4 v5 v5 v6 v6 v7 v7 v8 v8 v9 v9 H2: G can be decomposed into H1, H2.  = {H1, H2} is a packing of G.

11. v1 v2 v3 H1: v4 v5 v6 v7 v8 v9 v1 v2 v3 H2: v4 v5 v6 v7 v8 v9 • ‘= {(v1, v5, v3, v6), (v1, v2, v5, v4), (v1, v7, v2, v9), (v2, v3, v7, v6), • (v2, v4, v3, v8), (v1, v3, v9, v8)} is a 4-cycle packing of G. G has a 4-cycle system.

12. Cycle Decomposition Alspach Conjecture : Let 3 m1, m2, ..., mtn such that m1 + m2 + ... + mt = n(n–1)/2 for odd n (m1 + m2 + ... + mt = n(n–2)/2 for even n). Then Kn(Kn – F) can be decomposed into cycles C1, C2, ..., Ct such that Ci is a mi-cycle for i = 1, 2, ..., t. D. Bryant, D. Horsley and W. Pettersson, Cycle decompositions V: Complete graphs into cycles of arbitrary lengths, arXiv:1204.3709v2 [math.CO], 2013.

13. Cycle Decomposition D. Sotteau, Decomposition of Km,n(Km,n*) into cycles (circuits) of length 2k, J. Combin. Theory B, 30 (1981) 75.81. Theorem 1: There exists a 2k-cycle decomposition of Km,nif and only if each vertex has even degree, mn is divisible by 2k, and m, n k.

14. Does there exist a 4-cycle system of Kn – E(G) for any 4-regular subgraphG of Kn?

15. Known Results

16. Known Results A. Kotzig, On decomposition of the complete graph into 4k-gons, Mat.-Fyz. Cas., 15 (1965), 227-233. Theorem 2: There exists a 4-cycle system of Knif and only if n ≡ 1 (mod 8).

17. Known Results B. Alspach and S. Marshall, Even cycle decompositions of complete graphs minus a 1-factor, J. Combin. Des., 2 (1994), 441-458. Theorem 3: There exists a 4-cycle system on Kn – F, where F is a 1-factor of Kn, if and only if n ≡ 0 (mod 2).

18. Known Results H.-L. Fu and C. A. Rodger, Four-Cycle Systems with Two- Regular Leaves, Graphs and Comb., 17 (2001), 457-461. Theorem 4: Let F be a 2-regular subgraph of Kn. There exists a 4-cycle system of Kn– F if and only if n is odd and 4 divides the number of edges of Kn– F.

19. Known Results C.-M. Fu, H.-L. Fu, C. A. Rodger and T. Smith, All graphs with Maximum degree three whose complements have 4-cycle Decompositions, Discrete Math., 308 (2008), 2901-2909. Theorem 5: Let G be a graph on n vertices, where n is even and (G)  3. Then there exists a 4-cycle system of Kn – E(G) if and only if (1) All vertices in G have odd degree, (2) 4 divides n(n–1)/2 – |E(G)|, and (3) G is not one of the two graphs of order 8 as follows.

20. Let G be a 4-regular subgraph of Kn.Does there exist a 4-cycle system of Kn – E(G)?

21. 4-Regular Graphs

22. Some 4-regular graphs

23. Question: Does there exist a 4-cycle system of Kn – E(K5) ? 1. n = 5, Yes! 2. n = 6, No! 3. n = 7, No! 4. n = ?, Yes!

24. Question: Does there exist a 4-cycle system of Kn – E(K5) ? n 5 is odd and 4 | n(n – 1) / 2 – 10 ⇒ 4 | (n2–n– 20) / 2 ⇒ 8 | (n –4)(n– 5) ⇒ n  5 (mod 8).

25. Question: Does there exist a 4-cycle system of Kn – E(K5) ? Answer: n  5 (mod 8), Yes! Let n = 8k + 5. K8k+5 – E(K5) = K8k+1K4, 8k. ... K8k+1 K4, 8k

26. Lemma 6: There exists a 4-cycle system of Kn – E(K5) if and only if n  5 (mod 8).

27. Question: Does there exist a 4-cycle system of Kn – E(G) ? n 6 is odd and 4 | n(n – 1) / 2 – 12 ⇒ 4 | n(n – 1) / 2 ⇒ 8 | n(n – 1) ⇒ n  1 (mod 8). G:

28. Question: Does there exist a 4-cycle system of K9 – E(G) ? G: K9 – E(G) :

29. Question: Does there exist a 4-cycle system of K9 – E(G) ? Answer: Yes !

30. Question: Does there exist a 4-cycle system of Kn – E(G) ? Answer: n  1 (mod 8), Yes ! Let n = 8k + 1. Kn – E(G) = (K9 – E(G)) K8k–8K8k–8,9 = (K9 – E(G)) K8k–7K8k–8,8 G: Kn – E(G) K9 – E(G) K8k–8 G

31. Lemma 7: There exists a 4-cycle system of Kn – E(G) if and only if n  1 (mod 8). G:

32. Question: Does there exist a 4-cycle system of Kn – E(G)? n 6 is odd and 4 | n(n – 1) / 2 – 12 ⇒ 4 | n(n – 1) / 2 ⇒ 8 | n(n – 1) ⇒ n  1 (mod 8). G:

33. Question: Does there exist a 4-cycle system of K9 – E(G)? Answer: No! K9 – E(G) : G: K9 – E(G) :

34. Lemma 8: There exists a 4-cycle system of Kn – E(G) if and only ifn  1 (mod 8) and n  17. G:

35. Q(t) = {G | G is any connected 4-regular graph with t vertices}.

36. s-reducible Definition 9: Let G be a 4-regular graph of order t. If there exists SV(G), |S| = s and a graph H where V(H) = N1(S) and E(H) E(G) = ∅ such that (G –S) H is 4-regular, then we call the graph G is s-reducible.

37. S = {∞1} V(H) = N1(S) = {v1, v2, v3, v4} E(H) = {v1v4, v2v3} ∞1 v1 v2 v1 v1 v2 v2 G: G – S: v3 v4 v3 v3 v4 v4 v5 v5 v6 v6 v5 v6 (G – S) H: G is 1-reducible.

38. 3-reducible Theorem 10: Let t  8 and G be a 4-regular graph of order t. If G contains a component with at least 6 vertices, then G is 3-reducible.

39. Sufficient Condition Theorem 11: Let G be a 4-regular of order t. If there exists a 4-cycle system of Kn– E(G), then (1) n≣ 1 (mod 8), for t is even and (2) n≣ 5 (mod 8), for t is odd.

40. Construction n≣ 1 (mod 8), t is even. n≣ 5 (mod 8), t is odd. G is 3-reducible 4-regular graph of order t. Kn– E(G) = [Kn–4– E((G – S)  H)]  R. Kn – E(G) (G – S) H H Kn–4– E((G – S)  H) S

41. Main Results

42. Main Theorem: Let G be any 4-regular graph with t vertices. There exists a 4-cycle system of Kn – E(G), if n is odd, 4 | n(n – 1)/2 – 2t, and (1) G is a vertex-disjoint union of t/5 copies of K5. (2) n (4t – 5)/3. (3) n > 9 for the following two graphs.

43. Future Work

44. Question 1. Let G be a 4-regular graph of order t. Does there exist a 4-cycle system of Kn– E(G) for tn < (4t – 5)/3? Question 2. Let G be a 4-regular graph of order t and t≣ 5 (mod 8). Is G 5-reducible? Question 3. Let G be a 4-regular spanning subgraph of Kn. Does there exist a 4-cycle system of Kn– E(G) for n ≡ 5 (mod 8)?

45. Thanks for your patient.