Three-matching intersection conjecture for perfect matching polytopes of small dimensions

Abstract In the study of Fulkerson’s conjecture, Fan and Raspaud (1994) proposed a weaker conjecture: in every bridgeless cubic graph G , there are three perfect matchings M 1 , M 2 , and M 3 such that M 1 ∩ M 2 ∩ M 3 = 0 . In this note we prove that this conjecture is true if the dimension of the perfect matching polytope of G is at most 9. This includes the class of bridgeless cubic graphs with the property that all the vertices of the corresponding perfect matching polytope are affinely independent.