Partitions of some planar graphs into two linear forests

A linear forest is a forest in which every component is a path. It is known that the set of vertices V (G) of any outerplanar graph G can be partitioned into two disjoint subsets V1, V2 such that induced subgraphs 〈V1〉 and 〈V2〉 are linear forests (we say G has an (LF ,LF)partition). In this paper, we present an extension of the above result to the class of planar graphs with a given number of internal vertices (i.e., vertices that do not belong to the external face at a certain fixed embedding of the graph G in the plane). We prove that there exists an (LF ,LF)-partition for any plane graph G when certain conditions on the degree of the internal vertices and their neighbourhoods are satisfied.