Planar k-cycle resonant graphs with k=1, 2

A connected graph is said to be k-cycle resonant if, for 16 t6 k, any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M -alternating cycle. The concept of k-cycle resonant graphs was introduced by the present authors in 1994. Some necessary and su6cient conditions for a graph to be k-cycle resonant were also given. In this paper, we improve the proof of the necessary and su6cient conditions for a graph to be k-cycle resonant, and further investigate planar k-cycle resonant graphs with k =1; 2. Some new necessary and su6cient conditions for a planar graph to be 1-cycle resonant and 2-cycle resonant are established. ? 2003 Elsevier B.V. All rights reserved.