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M-alternating Hamilton paths and M-alternating Hamilton cycles

M-alternating Hamilton paths and M-alternating Hamilton cycles. 张赞波 中山大学计算机科学系 广东轻工职业技术学院计算机工程系. Note. The material presented here is based on the submitted paper: Li, Lou & Zhang. M-alternating Hamilton paths and M-alternating Hamilton cycles. Outline. Definition & introduction

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M-alternating Hamilton paths and M-alternating Hamilton cycles

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  1. M-alternating Hamilton paths and M-alternating Hamilton cycles 张赞波 中山大学计算机科学系 广东轻工职业技术学院计算机工程系

  2. Note • The material presented here is based on the submitted paper: Li, Lou & Zhang. M-alternating Hamilton paths and M-alternating Hamilton cycles

  3. Outline • Definition & introduction • M-alternating cycles in bipartite graphs • M-alternating paths in general graphs • M-alternating cycles in general graphs • Summary

  4. Outline • Definition & introduction • M-alternating cycles in bipartite graphs • M-alternating paths in general graphs • M-alternating cycles in general graphs • Summary

  5. M-alternating paths (cycles) • For a graph G and a matching M, M-alternating paths (cycles) is the paths (cycles) of which the edges appear alternately in M and E(G)\M.

  6. A-type M-alternating paths • For convenience we call an edge in M or a M-alternating path starting and ending with edges in M an A-type M-alternating path.

  7. k-extendable • G is k-extendable is G has a P. M. and every matching in G of size 0  k  (υ-2)/2 is contained in a P. M.. A 1-extendable graph: prism

  8. The role of M-alternating paths/cycles • Tutte’s augmenting path theorem. • (Aldred, Holton, Lou, Saito, 2003) Let G(U,W) be a k-extendable bipartite graph and x ∈ U and y ∈ W, then there are k internally disjoint M-alternating paths between x and y. • …

  9. Extendablity and connectivity • Lemma (Plummer, 1980). If G is a k-extendable graph then κ k+1. • Lemma (Lou & Yu, 2004). If G is a k-extendable graph with k υ/4 then either G is bipartite or κ 2k.

  10. Lou’s conjecture • Conjecture (Lou) If G is a k-extendable graph with kυ/4 then G has an M-alternating Hamilton cycle for every perfect matching M. Note that here it is obviously that such a graph G has a Hamilton cycle. Since this kind of graph is rather dense, Lou raised the conjecture. But our results turn out that the existing of M-alternating Hamilton cycles almost decided only by the connectivity.

  11. Proof sketch • If G is k-ext where k υ/4, then • If G is bipartite, κ k+1 υ/4+1; • If G is general, κ 2k υ/2; • If G is bipartite and κυ/4+1, then there exists an M-alternating Hamilton cycle for every P. M. M. • If κυ/2, then there exists an M-alternating Hamilton cycle for every P. M. M. (Exception here!)

  12. Outline • Definition & introduction • M-alternating cycles in bipartite graphs • M-alternating paths in general graphs • M-alternating cycles in general graphs • Summary

  13. Main Result • Theorem 2.1. Let G be a bipartite graph with κυ/4+1 and M a P. M. of G. Then G has an M-alternating Hamilton cycle.

  14. Proof of the main result By contradiction. Suppose that C is the longest M-alternating cycle in G but is not hamiltonian. And let P=v0v1…v2p-1 be the longest A-type M-alternating path in G-C. Then the neighbors of v0 and v2p-1 are on P and C only. Count the degree sum of v0 and v2p-1. v0 v2p+1 P G-C-P C

  15. Outline • Definition & introduction • M-alternating cycles in bipartite graphs • M-alternating paths in general graphs • M-alternating cycles in general graphs • Summary

  16. Main Result • Theorem 3.1. Let G be a graph with a P. M. M. For any x, yV(G) connected by an A-type M-alternating path, d(x)+d(y)  υ. Then G has an A-type M-alternating Hamilton path. To the proof of theorem 4.2

  17. Proof of the main result By contradiction. Suppose that P=v0v1…v2p-1 is the longest A-type M-alternating path in G but is not hamiltonian. Count the degree sum of v0 and v2p-1. v0 v2p+1 P Forbidden structures

  18. Outline • Definition & introduction • M-alternating cycles in bipartite graphs • M-alternating paths in general graphs • M-alternating cycles in general graphs • Summary

  19. Exceptional graph • We call a graph G X-type if it is constructed by taking two copies of the complete graph K2n+1, n  1, with vertex sets {x1, x2, …, x2n+1} and {y1, y2, …, y2n+1}, and joining every xi to yi, 0  i  2n+1. • The P. M. {xiyi, 0  i 2n+1} is called the jointing matching of G.

  20. Kn Kn x1 y1 x2 y2 … … x2n+1 y2n+1 Exceptional graph and the jointing matching

  21. Main Result • Theorem 4.2. Let G be a graph with κ υ/2 and M a P. M. of G. Then either G has an M-alternating Hamilton cycle or G is X-type and M is the jointing matching of G.

  22. Proof of the Main Result By contradiction. Let C be the longest M-alternating cycle in G but not hamitonian. Let G1=G-C. We prove that G1 satisfy the condition of theorem 3.1. So there is an A-type M-alternating Hamilton path P= v0v1…v2p-1 in G1. We discuss two cases. (Why is an A-type M-alternating path some important?) Case 1. There exist two vertices v2r and v2s+1 such that there is no A-type M-alternating Hamilton path in G1 connecting them. Case 2. For every two vertices v2r and v2s+1 there is an A-type M-alternating Hamilton path in G1 connecting them. In case 1, we count the degree sum of v2s and v2r+1. (Why consider v2s and v2r+1?) In case 2, we count the degree sum of V0={v2i} and V1={v2i+1}. In this case we regards V0 as one vertex and V1 as one vertex. In a sub case we find the X-type graph.

  23. G1 v0 v2p-1 P C u0 u1 G1 and C

  24. v0 v2p-1 P0 C u2i u2i+1 Why is an A-type M-alternating path so important? In the figure below, C=u0u0 …u2m-1 is the longest M-alternating cycle and P0 is an A-type M-alternating path. The edges v0u2i and v2p-1u2i+1 can not all be in E(G). So do v0u2i+1 and v2p-1u2i. And this is true for all 0 im. So the degree sum of v0 and v2p-1 is bounded.

  25. v2s v2r+1 P0 C u2i u2i+1 Why consider v2s and v2r+1? Firstly 2s+1<2r or v2r Pv2s+1 is the A-type M-alternating path connecting v2r and v2s+1. So v2sPv2r+1 is an A-type M-alternating path. And the edges from v2s and v2r+1 to C is “full”.

  26. Result on extendability • Corollary 4.3. Let G be a k-extendable graph with k υ/4, and M a P. M. of G. Then G has an M-alternating Hamilton cycle. Note here that the only thing we need to do is to prove the X-type graph is not k-extendable for k  υ/4!

  27. Outline • Definition & introduction • M-alternating cycles in bipartite graphs • M-alternating paths in general graphs • M-alternating cycles in general graphs • Summary

  28. What is interesting here?

  29. Comparison with some results on Hamiltonity • Bipartite graphs • (?) A bipartite graph has a Hamilton cycle if δυ/4+1. • A bipartite graph with a P. M. M has an M-alternating cycle if κ υ/4+1.

  30. Comparison with some results on Hamiltonity • General graphs • (Dirac, 1952) A graph Gwith υ 3 is hamiltonian if δ υ/2. • (Bondy & Chvátal, 1974) Let G be a simple graph and let u and v be nonadjacent vertices in G such that d(u)+d(v)  υ. Then G is hamiltonian if and only if G+uv is hamiltonian. • A graph with a P. M. M has an M-alternating cycle if κ υ/2, unless it is the X-type graph.

  31. Further consideration • Can we replace κ with δ? • Can we strengthen the result like Bondy & Chvátal did for Dirac’s result?

  32. One more consideration…between general graphs and bipartite graphs

  33. Properties between general and bipartite graphs • Let P(x) be an property of any simple graph such that x is an expression of some parameters of the graph (such as the matching number, connectivity, etc), under what condition any that • If P(x) holds in general graphs then P(x/2+1) holds in bipartite graphs? • If P(x/2+1) holds in bipartite graphs then P(x) holds for general graphs?

  34. Example 1: Hamiltonian • Let P(x) be the property that if δ x then G has a Hamilton cycle, then P(υ/2) holds for genenral graphs. And we know that P(υ/4+1) holds for bipartite graphs.

  35. Example 2: Extendability • Let P(x) be the property that if a graph G is k-extendable with k  υ/4 then κ(G) x. We know that P(2k) holds for general graph and P(k+1) holds for bipartite graph.

  36. (Counter-)Example 3:M-alternating Hamilton cycle • Let P(x) be the property that, if κ(G)  x then for any P. M. M of a graph G, there is an M-alternating Hamilton cycle in G. We know that P(x) does not hold for general graphs, for there exist the exceptional X-type graphs, but P(x/2+1) holds for bipartite graphs.

  37. Questions • Are there more such properties P(x)? • How can we summarize and describe such kind of properties?

  38. Thanks for your attention, and any suggestion or problem is welcome!

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