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Packing Graphs with 4-Cycles: Insights and Results from Combinatorial Research

This paper explores the relationship between graph structures and the existence of cycle packings, specifically focusing on 4-cycles in graphs. We define key concepts such as regular graphs and cycle decompositions, and summarize known results. Our main findings address the conditions under which a 4-cycle system can exist within complete graphs and their subgraphs, presenting notable theorems related to graph regularity. Future work will expand on these methods to uncover deeper graph packing potential, contributing to the field of combinatorial mathematics.

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Packing Graphs with 4-Cycles: Insights and Results from Combinatorial Research

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  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.

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