Even Cycles And Perfect Matchings Of Claw-free Cubic Graphs

Let G be a graph and C be a cycle of G.If G-V(C)has a perfect matching,then C is called a nice cycle of G.If G-V(C)has a unique perfect matching,then C is called a forced cycle of G.If each even cycle in G is a nice cycle,then G is called a cycle-nice graph.If each induced even cycle in G is a nice cycle,then G is called an induced-cycle-nice graph.If each even cycle in G is a forced cycle,then G is called a cycle-forced graph.If each induced even cycle in G is a forced cycle,then G is called an induced-cycle-forced graph.A graph G is PM-compact if for each even cycle C of G,G-V(C)has at most one perfect matching.If G is PM-compact but not cycle-forced,then G is said a cycle-bad graph.For PM-compact graphs,there are some good results,including the complete characterizations of bipartite graphs and near-bipartite graphs,as well as claw-free cubic graphs.For induced-cycle-nice graphs and cycle-forced graphs,some results are obtained,including the complete characterizations of cycle-forced hamiltonian bipartite graphs and bipartite graphs;the complete characterization of induced-cycle-nice 2-connected claw-free cubic graphs,and the degree conditions of a graph to be an induced-cycle-nice graph.In this paper,we study induced-cycle-forced graphs,cycle-forced graphs and cycle-nice graphs.The main results are the following:●A complete characterization of induced-cycle-forced 2-connected claw-free cubic graphs.●A complete characterization of cycle-forced 2-connected claw-free cubic graphs.●A complete characterization of cycle-nice 2-connected claw-free cubic graphs.