Decomposition of triangle-free planar graphs

Abstract

A decomposition of a graph $G$ is a family of subgraphs of $G$ whose edge sets form a partition of $E\left(G\right)$. 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 Škrekovski (1999) that every triangle-free planar graph is 1-defective 3-choosable.