Alternating Perfect Matchings In Cubic Graphs

A perfect matching in a graph G is a set of pairwise nonadjacent edges which covers all vertices of G.In 1994,Fan and Raspaud proposed a conjecture that every bridgeless cubic graph contains three perfect matchings with empty intersection.We call this conjecture Fan-conjecture.Three perfect matchings with empty intersection in Fan-conjecture are called alternating perfect matchings.In this paper,we study this conjecture on some special classes of graphs.For cubic bipartite graphs,Fan-conjecture holds.A near-bipartite graph G is a nonbipartite matching covered graph with two distinct edges e1 and e2 such that G-e1-e2 is a matching covered bipartite graph.This is one class of graphs considered in this thesis.Let F be a 2-factor of a cubic graph G which consists of four cycles,and let C be a cycle of F.We refer C as a central cycle of G if each edge of E(G)\E(F)has one end in C.Jinkai Chen showed that Fan-conjecture holds for bridgeless cubic graphs each of which has a central cycle with no chords.This thesis consider bridgeless cubic graphs each of which has a central cycle with only one chord.A graph is essentially 4-edge-connected if it is 2-edge-connected and all 3-cut are trivial.Some essentially 4-edge-connected graphs with given properties are considered in this thesis.The main results of this thesis are as follows:(1)Every near-bipartite cubic graph has pairwise disjoint alternating perfect matchings.(2)If G is a bridgeless cubic graph which has a 2-factor consisting of four cycles one of which is a central cycle with only one chord,then G has alternating perfect matchings.(3)For some essentially 4-edge-connected cubic bricks,we obtain partial characterizations and prove the existence of alternating perfect matchings.