Linear arboricity of degenerate graphs

Abstract

A linear forest is a union of vertex‐disjoint paths, and the linear arboricity of a graph , denoted by , is the minimum number of linear forests into which the edge set of can be partitioned. Clearly, for a graph with maximum degree . On the other hand, the Linear Arboricity Conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that for every graph . This conjecture has been verified for planar graphs and graphs whose maximum degree is at most , or is equal to or . Given a positive integer , a graph is ‐degenerate if it can be reduced to a trivial graph by successive removal of vertices with a degree at most . We prove that for any ‐degenerate graph provided .