The three 4-flows conjecture
Status open medium confidence
The three 4-flows conjecture, which asks for a partition of the edge set of every bridgeless graph into three sets each of which can be removed to leave a graph with a nowhere-zero 4-flow, remains open. No post-2007 paper resolving or substantially partially resolving this conjecture was found; the problem reduces to cubic snarks and is implied by the Petersen coloring conjecture, itself still open.
Reviewer notes. The OPG page (openproblemgarden.org) returned ECONNREFUSED. Five search queries were run; none returned a paper specifically addressing the three 4-flows conjecture. The arXiv paper 2511.01556 (Mattiolo, Nov 2025) on removable edge subsets in graphs with nowhere-zero 4-flows is related but does not address the conjecture directly. The Petersen coloring conjecture (which implies the three 4-flows conjecture) and the orientable cycle four cover conjecture (which is implied by it) are both still open, confirming the problem likely remains unresolved. Confidence is medium rather than high because the specific conjecture did not appear in any indexed paper by name, making it difficult to rule out niche partial results.
Discussion
A graph $ G $ has a nowhere-zero 4-flow if and only if there exist disjoint sets $ A_1,A_2,A_3 \subseteq E(G) $ with $ A_1 \cup A_2 \cup A_3 = E(G) $ so that $ G\A_i $ has a nowhere-zero 2-flow for $ 1 \le i \le 3 $ . 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 $ G $ with no cut-edge, there exist disjoint sets $ A_B_1,B_2 \subseteq E(G) $ with $ A \cup B_1 \cup B_2 = E(G) $ and so that $ G\B_1 $ and $ G\B_2 $ have nowhere-zero 3-flows and $ G\A $ has a nowhere-zero 2-flow. The Petersen graph has such a decomposition ( $ B_1 $ and $ B_2 $ 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.
Bibliography
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[J]F. Jaeger, On circular flows in graphs. Finite and infinite sets, Vol. I, II (Eger, 1981), 391--402, Colloq. Math. Soc. János Bolyai, 37, North-Holland, Amsterdam, 1984.. MathSciNet MathSciNet