Edge disjoint cycles through specified vertices

We give a sufficient condition for a simple graph G to have k pairwise edge-disjoint cycles, each of which contains a prescribed set W of vertices. The condition is that the induced subgraph G[W ] be 2k-connected, and that for any two vertices at distance two in G[W ], at least one of the two has degree at least |V (G)|/2 + 2(k − 1) in G. This is a common generalization of special cases previously obtained by Bollobás/Brightwell (where k = 1) and Li (where W = V (G)). A key lemma is of independent interest. Let G be the complement of a bipartite graph with partite sets X , Y . If G is 2k connected, then G contains k Hamilton cycles which are pairwise edge-disjoint except for edges in G[Y ].