The Tutte 4-flow conjecture asserts that every bridgeless graph having no Petersen graph as a graph minor admits a nowhere-zero flow whose nonzero integer values on its edges have absolute value less than 4 (Tutte 1966, Mkrtchyan 2020). For cubic graphs, existence of such a flow is equivalent to edge chromatic number 3, so the Tutte three-edge-coloring conjecture is the cubic-graph special case (Inoue et al. 2026).
Tutte 4-Flow Conjecture
See also
Flow Polynomial, Nowhere-Zero Flow, Petersen Graph, Tutte Conjecture, Tutte Three-Edge-Coloring ConjectureExplore with Wolfram|Alpha
References
Inoue, Y.; Kawarabayashi, K.; Matsuo, R.; Miyashita, A.; Mohar, B.; and Sonobe, T. "Three-Edge-Coloring Apex Cubic Graphs." 24 Aug 2026. https://arxiv.org/abs/2608.22870.Mkrtchyan, V. "Sublinear Bounds for Nullity of Flows and Approximating Tutte's Flow Conjectures." 17 Aug 2020. https://arxiv.org/abs/2008.07152.Tutte, W. T. "On the Algebraic Theory of Graph Colorings." J. Combin. Th. 1, 15-50, 1966.Cite this as:
Weisstein, Eric W. "Tutte 4-Flow Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Tutte4-FlowConjecture.html