Long cycles in unbalanced bipartite graphs

Let G[X,Y ] be a 2-connected bipartite graph with |X| ≥ |Y |. For S ⊆ V (G), we define δ(S;G) := min{dG(v) : v ∈ S}. We define σ1,1(G) := min{dG(x) + dG(y) : x ∈ X, y ∈ Y, xy / ∈ E(G)} and σ2(X) := min{dG(x) + dG(y) : x, y ∈ X,x 6= y}. We denote by c(G) the length of a longest cycle in G. Jackson [J. Combin. Theory Ser. B 38 (1985), 118–131] proved that c(G) ≥ min{2δ(X;G) + 2δ(Y ;G)− 2, 4δ(X;G)− 4, 2|Y |}. In this paper, we extend this result, and prove that c(G) ≥ min{2σ1,1(G)− 2, 2σ2(X)− 4, 2|Y |}.