Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths

TL;DR

This paper proves that planar graphs with girth at least 6 can be 2-colored so that each monochromatic component is a short path, extending previous results for higher girth and providing counterexamples for girth 4.

Contribution

It introduces a new coloring result for planar graphs of girth 6, ensuring short monochromatic paths, and constructs graphs showing limitations for girth 4.

Findings

Planar graphs of girth ≥6 can be 2-colored with short monochromatic paths

A list coloring version of the main result is established

Counterexamples show long monochromatic paths in girth 4 graphs

Abstract

Recently, Borodin, Kostochka, and Yancey (On $1$-improper $2$-coloring of sparse graphs. Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least $7$ can be $2$-colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least $6$ admits a vertex coloring in $2$ colors such that each monochromatic component is a path of length at most $14$. Moreover, we show a list version of this result. On the other hand, for each positive integer $t\geq 3$, we construct a planar graph of girth $4$ such that in any coloring of vertices in $2$ colors there is a monochromatic path of length at least $t$. It remains open whether each planar graph of girth $5$ admits a $2$-coloring with no long monochromatic paths.