Path partition of planar graphs with girth at least six

Abstract

A graph [Formula: see text] has a [Formula: see text]-partition if [Formula: see text] can be partitioned into two nonempty disjoint subsets [Formula: see text] and [Formula: see text] so that [Formula: see text] and [Formula: see text] are graphs whose components are paths of order at most [Formula: see text] and [Formula: see text], respectively. In this paper, we proved that every planar graph with girth at least six giving that [Formula: see text]-cycle is not intersecting with [Formula: see text]-cycle admits a [Formula: see text]-partition, where [Formula: see text] and [Formula: see text].