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arXiv · 2609.19503

Total-coloring of planar graphs with maximum degree 6 and without prescribed 4-cycles

Enqiang Zhu·Yangyang Zhou·Jin Xu
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Abstract

The Total Coloring Conjecture (TCC) is a challenging unsolved problem posed by Behzad and Vizing independently, which states that every simple graph $G$ admits a ($Δ(G)$ +2)-total-coloring, where $Δ(G)$ denotes the maximum degree of $G$. This conjecture has been confirmed for graphs with $Δ(G)\leq 5$. However, for planar graphs, the only open case is $Δ(G)=6$. It was known that planar graphs with maximum degree 6 and without 4-cycles are 7-totally-colorable. In this paper, we improve this result by showing that any planar graph $G$ of maximum degree 6, which does not contain some special 4-cycles, is 7-totally-colorable.

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Enqiang Zhu, Yangyang Zhou, Jin Xu. 2026-09-16. Total-coloring of planar graphs with maximum degree 6 and without prescribed 4-cycles. https://arxiv.org/abs/2609.19503

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