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 { d G ( v ) : v ∈ S } . We define σ 1 , 1 ( G ) : = min { d G ( x ) + d G ( y ) : x ∈ X , y ∈ Y , x y ∉ E ( G ) } and σ 2 ( X ) : = min { d G ( x ) + d G ( y ) : x , y ∈ X , x ≠ 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 δ ( X ; G ) + 2 δ ( Y ; G ) − 2 , 4 δ ( X ; G ) − 4 , 2 | Y | } . In this paper, we extend this result, and prove that c ( G ) ≥ min { 2 σ 1 , 1 ( G ) − 2 , 2 σ 2 ( X ) − 4 , 2 | Y | } .