Planar graphs without short even cycles are near-bipartite

Abstract

A graph is {\em near-bipartite} if its vertex set can be partitioned into an independent set and a set that induces a forest. It is clear that near-bipartite graphs are $3$-colorable. In this note, we show that planar graphs without cycles of lengths in $\{4,6,8\}$ are near-bipartite.