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

Authors:
Xiaofeng Guo;Fuji Zhang
Affiliations:
Department of Mathematics, Xiamen University, Xiamen Fujian 361005, People's Republic of China and Institute of Mathematics, Xinjiang University, Wulumuqi Xinjiang 830046, People's Republic of Chi ...;Department of Mathematics, Xiamen University, Xiamen Fujian 361005, People's Republic of China
Venue:
Discrete Applied Mathematics
Year:
2003

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Abstract

A connected graph is said to be k-cycle resonant if, for 1 ≤ t ≤ 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 sufficient conditions for a graph to be k-cycle resonant were also given. In this paper, we improve the proof of the necessary and sufficient 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 sufficient conditions for a planar graph to be 1-cycle resonant and 2-cycle resonant are established.