  In connection with Fulkerson's conjecture on cycle covers, Fan and Raspaud
proposed a weaker conjecture: For every bridgeless cubic graph $G$, there are
three perfect matchings $M_1$, $M_2$, and $M_3$ such that $M_1\cap M_2 \cap
M_3=\emptyset$. We call the property specified in this conjecture the three
matching intersection property (and 3PM property for short). We study this
property on matching covered graphs. The main results are a necessary and
sufficient condition and its applications to characterization of special
graphs, such as the Halin graphs and 4-regular graphs.
