LONGEST CYCLES IN 2-CONNECTED BIPARTITE GRAPHS

Let G be a 2-connected bipartite graph with bipartition (A_1,A_2).If d(u,v)=2 and max {d(u),d(v)}≥δ~* for any u,v∈V (G),then G contains a cycleof length at least 2 min (|A_1|,|A_2|,2δ~*-2).Moreover,if |A_2|=|A_2|,δ~*≥1/2(|A_1|+1),then G has a Hamiltonian cycle.