Year: 2026
Type: preprint
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 Kardoš's Hamiltonicity theorem for three-connected cubic planar graphs.
Source: arXiv (Cornell University)
Author: Zhu, Enqiang
Institution: Guangzhou University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223700231