The Borowiecki-Broere Generalized Total Coloring Conjecture for Planar Graphs

James Alexander Schreib

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arXiv · 2610.00091

The Borowiecki-Broere Generalized Total Coloring Conjecture for Planar Graphs

James Alexander Schreib

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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.

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James Alexander Schreib. 2026-09-07. The Borowiecki-Broere Generalized Total Coloring Conjecture for Planar Graphs. https://arxiv.org/abs/2610.00091

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