Open neighbourhood colouring of some path related graphs

Abstract

An open neighbourhood \(k\)-colouring of a simple connected undirected graph \(G(V, E)\) is a \(k\)-colouring \(c : V \rightarrow \{1, 2, \dots, k\}\), such that, for every \(w \in V\) and for all \(u, v \in N(w)\), \(c(u) e c(v)\). The minimal value of \(k\) for which \(G\) admits an open neighbourhood \(k\)-colouring is called the open neighbourhood chromatic number of \(G\) and is denoted by \(\chi_{onc}(G)\). In this paper, we obtain the open neighbourhood chromatic number of the line graph and total graph of a path \(P_n\). We also obtain the open neighbourhood chromatic number of two families of graphs which are derived from a path \(P_n\), namely \(k\)th power of a path and transformation graph of a path.