The Vertex Linear Arboricity of a Special Class of Integer Distance Graphs

Abstract

The vertex linear arboricity \(vla(G)\) of a nonempty 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.An integer distance graph is a graph \(G(D)\) with the set of all integers as vertex set and two vertices \(u,v \in {Z}\) are adjacent if and only if \(|u-v| \in D\), where the distance set \(D\) is a subset of the positive integers.Let \(D_{m,k,3} = [1,m] \setminus \{k, 2k, 3k\}\) for \(m \geq 4k \geq 4\). In this paper, we obtain some upper and lower bounds of the vertex linear arboricity of the integer distance graph \(G(D_{m,k,3})\) and the exact value of it for some special cases.