Decomposition of triangle-free planar graphs

TL;DR

This paper proves that every triangle-free planar graph can be decomposed into a 2-degenerate graph and a matching, leading to new coloring properties and strengthening previous results on list colorings.

Contribution

It introduces a novel decomposition of triangle-free planar graphs into a 2-degenerate graph and a matching, advancing understanding of their coloring properties.

Findings

Every triangle-free planar graph decomposes into a 2-degenerate graph and a matching.

Such graphs have a matching making the remaining graph online 3-DP-colorable.

This result strengthens earlier theorems on defective list colorings.

Abstract

A decomposition of a graph $G$ is a family of subgraphs of $G$ whose edge sets form a partition of $E(G)$. In this paper, we prove that every triangle-free planar graph $G$ can be decomposed into a $2$-degenerate graph and a matching. Consequently, every triangle-free planar graph $G$ has a matching $M$ such that $G-M$ is online 3-DP-colorable. This strengthens an earlier result in [R. \v{S}krekovski, {\em A Gr\"{o}tzsch-Type Theorem for List Colourings with Impropriety One}, Combin. Prob. Comput. 8 (1999), 493-507] that every triangle-free planar graph is $1$-defective $3$-choosable.