Year: 2026
Type: preprint
Abstract: We prove the conjecture of Borowiecki and Broere that every finite simple planar graph admits a generalized total coloring with four colors: a proper vertex coloring with four colors together with an edge coloring with four colors in which each edge color class is a forest and no edge receives the color of either endpoint. In fact every proper four-coloring of such a graph extends to an edge coloring of the required kind. The proof combines a component-count inequality for planar graphs whose vertices are partitioned into four independent sets, obtained by completing to a triangulation, with the matroid partition theorem applied to four graphic matroids. A Lean 4 formalization relative to six named background assumptions is described below.
Source: arXiv (Cornell University)
Author: Schreib, James Alexander
Institution: New York University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7216146432