Long cycles in unbalanced bipartite graphs

Let G vertical bar X, Y vertical bar be a 2-connected bipartite graph with vertical bar X vertical bar >= vertical bar Y vertical bar. For S subset of V(G), we define delta(S; G) := min{d(G)(nu) : nu is an element of S}. We define 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)} and sigma(2)(X) := min{d(G)(x) + d(G)(y) : x, y is an element of X, x not equal y}. We denote by c(G) the length of a longest cycle in G. Jackson [B. Jackson, Long cycles in bipartite graphs, J. Combin. Theory Ser. B 38 (1985) 118-131] proved that c(G) >= min{2 delta(X; G) + 2 delta(Y; G) - 2, 4 delta(X; G) - 4, 2 vertical bar Y vertical bar}. In this paper, we extend this result, and prove that c(G) >= min{2 sigma(1,1)(G) - 2, 2 sigma(2)(X) - 4, 2 vertical bar Y vertical bar}. (C) 2012 Elsevier B.V. All rights reserved.