Open Problem Garden

A graph has a nowhere-zero 4-flow if and only if there exist disjoint sets with so that has a nowhere-zero 2-flow for . Thus, the above conjecture is true with room to spare for such graphs. Since every 4-edge-connected graph and every 3-edge-colorable cubic graph has a nowhere-zero 4-flow, this conjecture is automatically true for these families. As with the 5-flow conjecture or the cycle double cover conjecture, establishing this conjecture comes down to proving it for cubic graphs which are not 3-edge-colorable.

This conjecture is a consequence of the Petersen coloring conjecture, and it implies the Orientable cycle four cover conjecture. The latter implication follows immediately from the fact that every graph with a nowhere-zero 4-flow has an orientable cycle double cover. Actually, it is possible that for every graph with no cut-edge, there exist disjoint sets with and so that and have nowhere-zero 3-flows and has a nowhere-zero 2-flow. The Petersen graph has such a decomposition ( and should be alternate edges of some 8-circuit) and so does every graph with a nowhere-zero 4-flow. If this stronger statement is true, then it would imply the oriented eight cycle four cover conjecture.