Hamiltonian and long cycles in bipartite graphs with connectivity

Let G be a graph,.(G) the order of G, kappa(G) the connectivity of G and k a positive integer such that k <= (nu(G) - 2)/2. Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A graph G is Hamiltonian if it contains a Hamiltonian cycle. In this paper, we define a parameter c(G) and show some of its properties in bipartite graphs. Let G be a bipartite graph with bipartition (X, Y) such that vertical bar X vertical bar = vertical bar Y vertical bar and c(G) > 0, and C a longest cycle in G. We show that 0 < c(G) <= kappa(G)- 1, G is Hamiltonian if nu(G) <= 2 kappa(G)+ 4c(G)-2, and vertical bar V(C)vertical bar >= 6c(G) if nu(G) > 6c(G). We show some of its corollaries in k-extendable, bipartite graphs and two conjectures in k-extendable graphs. (C) 2021 Elsevier B.V. All rights reserved.