Fan Conjecture Research In Special Cubic Graphs

In 1971,Fulkerson proposed a conjecture that each bridgeless cubic graph contain six perfect matchings,so that each edge is contained in exactly two of them.This conjecture is called Fulkerson Conjecture.In 1994,Fan and Raspaud proposed a weaker conjecture:each bridgeless cubic graph contains three perfect matchings with no edge in common.This conjecture is called Fan Conjecture.For a bridgeless cubic graph,if Fulkerson Conjecture holds then Fan conjecture holds.Both of these two conjectures are studied on some special graphs.Especially,in 2014,Macajova and Skoviera proved that for every bridgeless cubic graph which has a 2-factor with at most two odd circuits,Fan Conjecture holds.In 2002,Song proved that Fan Conjecture holds for bridgeless cubic graphs which have 2-factors with(1)at most two odd circuits and at most three even circuits,or(2)with four odd circuits,no even circuits and cyclically 4-edge connected.This paper studies Fan Conjecture for some special graphs.The main results are the following:·Fan Conjecture holds for bridgeless cubic graphs which have a 2-factor with only four odd circuits,one of which has no chords.·A sufficient and necessary condition for the existence of three perfect matchings with no edge in common in bridgeless cubic graphs.·Fan Conjecture holds for 1-Hamilton cubic graphs and 2-Hamilton cubic graphs·Fulkerson Conjecture and so Fan conjecture holds for two special classes of graphs Gt(m)and Gt(m,n).