Planar graphs without short even cycles are near-bipartite

TL;DR

This paper proves that planar graphs lacking cycles of lengths 4, 6, and 8 can be partitioned into an independent set and a forest, making them near-bipartite and 3-colorable.

Contribution

It establishes a new class of planar graphs without certain short even cycles are near-bipartite, expanding understanding of graph colorability.

Findings

Planar graphs without 4, 6, 8 cycles are near-bipartite.

Such graphs are 3-colorable.

The result broadens the class of graphs known to be 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.