Perfect Matching Covers For Bridgeless Cubic Graphs Of Oddness 4

Berge proposed a conjecture that there are five perfect matchings in every bridgeless cubic graph such that every edge in the graph is contained in at least one of these perfect matchings.According to the existing conclusions,Berge Conjecture is still a challenging problem in graph theory.We have turned attention to the weaker problem:whether there exists an integer c such that every bridgeless cubic graph can be covered by at most c perfect matchings.The main problem presented in this paper is that whether there exists a constant number of perfect matchings covering all the edges in a bridgeless cubic graph G,in which the number of odd circuits is 4.In this paper,by constructing alternate paths,dividing the circle to be covered as one edge to construct the bridgeless cubic graph and constructing bridgeless cubic graph based on the properties of the cycle chain structure starting from odd circles,the following results are obtained:1.There are 41 perfect matching covers with no even circles in 2 factor of the bridgeless cubic graph of oddness 4.2.There are 53 perfect matching covers with a class of even circles in 2 factor of the bridgeless cubic graph of oddness 4.