Tutteʹs 3-flow Conjecture and Matchings in Bipartite Graphs

Tutteʹs 3-flow conjecture is restated as the problem of finding an orientation of the edges of a 4-edge-connected, 5-regular graph G, for which the out-flow at each vertex is +3 or -3. The induced equipartition of the vertices of G is called mod 3-orientable. We give necessary and sufficient conditions for the existence of mod 3-orientable equipartitions in general 5-regular graphs, in terms of (i) a perfect matching of a bipartite graph derived from the equipartition and (ii) the size of cuts in G.