Partitioning triangle-free planar graphs into a forest and a linear forest
Guanwu Liu, Rongxing Xu
TL;DR
This paper proves that every triangle-free planar graph can be partitioned into a forest and a linear forest (i.e., a forest where each component is a path), equivalently achieving a maximum degree of $2$ for one part. It achieves this via a strengthened boundary-precoloring framework: for a triangle-free plane graph $G$ with a boundary vertex $z$, boundary sets $P$ (with $|P|\le 3$ and consecutive on the boundary) and an independent set $Q$, there exists a coloring $\phi: V(G)\to\{1,2\}$ extending a precoloring $\eta: P\to\{1,2\}$ and coloring $Q$ with $2$, such that color $1$ forms a linear forest and color $2$ forms a forest, together with structural conditions (C1)–(C5). The proof proceeds by a minimal counterexample and induction on the number of vertices, deriving key structural lemmas: $G$ is $2$-connected, the boundary cycle has length at least $5$, no separating $4$-cycles occur, and no boundary chords can arise; these enable successive extensions of partial colorings to the whole graph. The result improves prior bounds from $d\le 3$ to $d\le 2$ for the maximum degree in the second forest, contributing to the literature on acyclic colorings and vertex-partition decompositions in triangle-free planar graphs.
Abstract
Raspaud and Wang conjectured that every triangle-free planar graph can be vertex-partitioned into an independent set and a forest. Independently, Kawarabayashi and Thomassen also remarked that this might be true, after providing another proof of a result of Borodin and Glebov, showing this result for planar graphs of girth~5. Subsequently, Dross, Montassier, and Pinlou raised the same question and proved that every triangle-free planar graph can be partitioned into a forest and another forest of maximum degree~5. More recently, Feghali and Šámal improved this bound on the maximum degree to~3. In this note, we further improve the result by showing that every triangle-free planar graph can be partitioned into a forest and a linear forest, that is, a forest of maximum degree~2.