M-alternating Hamilton paths and M-alternating Hamilton cycles

We study M-alternating Hamilton paths, and M-alternating Hamilton cycles in a simple connected graph G on @n vertices with a perfect matching M. Let G be a bipartite graph, we prove that if for any two vertices x and y in different parts of G, d(x)+d(y)>[email protected]/2+2, then G has an M-alternating Hamilton cycle. For general graphs, a condition for the existence of an M-alternating Hamilton path starting and ending with edges in M is put forward. Then we prove that if @k(G)>[email protected]/2, where @k(G) denotes the connectivity of G, then G has an M-alternating Hamilton cycle or belongs to one class of exceptional graphs. Lou and Yu [D. Lou, Q. Yu, Connectivity of k-extendable graphs with large k, Discrete Appl. Math. 136 (2004) 55-61] have proved that every k-extendable graph H with k>[email protected]/4 is bipartite or satisfies @k(H)>=2k. Combining our result with theirs we obtain we prove the existence of M-alternating Hamilton cycles in H.