Hamilton cycles passing through a matching in a bipartite graph with high degree sum

Let G be a balanced bipartite graph of order 2n with partite sets X and Y, and M be a matching of k edges in G. Let sigma(1,1)(G) = min{d(G )(x) + d(G) (y) : x is an element of X, y is an element of Y, xy is not an element of /E(G)}. Conditions on sigma(1,1)(G) for the existence of a Hamilton cycle passing through M have been studied by many researchers, but the threshold is not known to be sharp for 2n/3 <= k <= n-2. In this paper we prove that G has a Hamilton cycle passing through all the edges of M if sigma(1,1)(G) >= 2n - k and k <= n - 2. Moreover, we show that the lower bound on sigma(1,1)(G) is sharp for every k with 2n/3 <= k <= n - 2.(c) 2023 Elsevier B.V. All rights reserved.