Partitioning planar graphs without 4-cycles and 6-cycles into a forest and a disjoint union of paths

TL;DR

This paper proves that planar graphs without 4- and 6-cycles can be partitioned into a forest and a disjoint union of paths, extending previous results on vertex partitions in such graphs.

Contribution

It introduces a new vertex partitioning method for planar graphs without 4- and 6-cycles, improving upon prior work by reducing the number of parts needed.

Findings

Every such planar graph can be partitioned into a forest and a disjoint union of paths.

This extends Wang and Xu's 2013 result on vertex partitions.

The partition ensures one part induces a forest and the other a forest with max degree 2.

Abstract

In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, where one induces a graph with maximum degree two, and the remaining two are independent sets.