Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Zhu, Enqiang, Zhou, Yangyang, Xu, Jin
Institutions: Guangzhou University, Peking University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213591991