The Vertex Linear Arboricity of Claw-Free Graphs with Small Degree

Abstract

The vertex linear arboricity \(vla(G)\) of a graph \(G\) is the minimum number of subsets into which the vertex set \(V(G)\) can be partitioned so that each subset induces a subgraph whose connected components are paths. It is proved here that \(\lceil \frac{\omega(G)}{2}\rceil \leq vla(G) \leq \lceil \frac{\omega(G)+1}{2}\rceil\) for a claw-free connected graph \(G\) having \(\Delta(G) \leq 6\), where \(\omega(G)\) is the clique number of \(G\).