Graph Theory » Coloring » Edge coloring

Black and White Cycle Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

Conjecture   Let $ G $ be a cubic graph with a nz-$ 4 $-flow, and let each edge $ e \in E(G) $ satisfying $ G-e $ does not have a nz-$ 4 $-flow be colored black, and let each edge $ e \in E(G) $ satisfying $ G-e $ has a nz-$ 4 $-flow be colored white. Then $ G $ contains a white cycle but not a black cycle.


Used terminology: nz-$ 4 $-flow is the abbrevation for nowhere-zero $ 4 $-flow, a cycle is a connected $ 2 $-regular graph, a cycle is called black (white) if each edge of the cycle is colored black (white).

Keywords: 3-edge-coloring; 4-flow; cubic graphs; removable edge

Posted by arthur
updated September 21st, 2026
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