Bipartite index of a graph and Vizing's theorem

For a graph  G the bipartite index is equal to the least number of edges
of G which removal yields a bipartite graph. We prove that if G is a
bridgeless cubic graph and the bipartite index bi(G) is at most two,
then G is 3-edge-colourable. The statement can be considered as a
generalisation of Hall's theorem for cubic graphs. The result is best
possible, since the flower snarks have bipartite index 3. Finally, we
will discuss possible generalisation of the statement to r-regular graphs.

Joint work with J. Karabas, E. Macajova and M. Skoviera.