Disjoint Even Cycles Packing

We generalize the well-known theorem of Corrádi and Hajnal which says that if a given graph G has at least 3k vertices and the minimum degree of G is at least 2k, then G contains k vertex-disjoint cycles. Our main result is the following; for any integer k, there is an absolute constant c k satisfying the following; let G be a graph with at least c k vertices such that the minimum degree of G is at least 2k. Then either (i) G contains k vertex-disjoint even cycles, or (ii) ( 2 k − 1 ) K 1 + p K 2 ⊆ G ⊆ K 2 k − 1 + p K 2 ( p ⩾ k ⩾ 2 ) , or k = 1 and each block in G is either a K 2 or a odd cycle, especially, each endblock in G is a odd cycle. In fact, our proof implies the following; the “even cycles” in the conclusion (i) can be replaced by “theta graphs”, where a theta graph is a graph that has two vertices x, y such that there are three disjoint paths between x and y. Let us observe that if there is a theta graph, then there is an even cycle in it. Furthermore, if the conclusion (ii) holds, clearly there are no k vertex-disjoint even cycles (and hence no k vertex-disjoint theta graphs).