Proof of a conjecture on cycles in a bipartite graph

Abstract

It was conjectured in [Wang, to appear in The Australasian Journal of Combinatorics] that, for each integer k ≥ 2, there exists M(k) such that if G= (V1, V2, E) is a bipartite graph with |V1| = |V2| = n ≥ M(k) and d(x) + d(y) ≥ n + k for each pair of nonadjacent vertices x and y of G with x ∈ V1 and y ∈ V2, then, for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck in G such that ei ∈ E(Ci for all i ∈ {1, …, k} and V(C1 ∪ ··· ∪ Ck) = V(G). This conjecture is also verified for k = 2, 3 in [Wang, to appear; Wang, manuscript]. In this article, we prove this conjecture to be true if n ≥ 3k, i.e., M(k) ≤ 3k. We will also show that, if n ≥ 3k, then for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck of length at most 6 in G such that ei ∈ E(Ci) for all i ∈ {1, …, k}. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 333–343, 1999