On Cycle-nice Graphs

An even cycle C in a graph G is called a nice cycle if G-V(C) has a perfect matching. A graph is called cycle-nice graph if its all even cycles are nice cycles.A graph is called induced-cycle-nice graph if its all induced even cycles are nice cycles; An even cycle of a graph G is a forcing cycle if G — V(C) has a unique perfect matching. A graph is called cycle-forced graph if its all even cycles are forcing cycles. A graph G is PM-compact if G — V(C) has at most one perfect matching for any even cycle C of G. By the definitions the relationship between the four kinds of graphs is as follows: if a graph is cycle-forced, then it is a special PM-compact graph and a special cycle-nice graph; if a graph is a cycle-nice graph, then it is an induced-cycle-nice graph. The converse is not true. For PM-compact graph, there are some good research results, including the complete characterization of PM-compact bipartite and near-bipartite graphs, as well as claw-free cubic graphs. For cycle-forced graph and induced-cycle-nice graph,there are some good results, including the complete characterization of cycle-forced hamiltonian bipartite graph, the partial characterization of cycle-forced bipartite graphs, and the degree conditions of a graph to be an induced-cycle-nice graph.In this paper, we study induced-cycle-nice graphs and cycle-forced bipartite graphs. The main results are the following:? A complete characterization of 2-edge-connected claw-free cubic graphs which are induced-cycle-nice graphs.? A complete characterization of near-hamiltonian bipartite graphs which are cycle-forced graphs.? Partial characterization of general bipartite graphs which are cycle-forced graphs.