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arXiv · 2610.32717

Total eight-colorings of maximal planar graphs of maximum degree six

Enqiang Zhu
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Abstract

We prove that every finite simple maximal planar graph of maximum degree at most six admits a total eight-coloring. We first show that every proper vertex four-coloring of a subcubic graph extends to a total five-coloring. A curvature inequality for boundary walks then gives a matching decomposition of four-connected triangulations with degrees five and six, extending their prescribed vertex four-colorings to total eight-colorings. In a minimum counterexample, two local boundary colorings force the degree-four vertices to have disjoint neighborhoods. A characterization of the local extension obstructions permits these vertices to be restored simultaneously. The proof uses Hamiltonian-connectedness of four-connected planar graphs and the computer-assisted Four-Color Theorem; it requires no further computation.

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Enqiang Zhu. 2026-10-23. Total eight-colorings of maximal planar graphs of maximum degree six. https://arxiv.org/abs/2610.32717

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