[r, s, t] - Colouring of Join Graphs S _n + O _m

Abstract

The concept of [r,s,t]-colourings was introduced by A. Kemnitz and M. Marangio in 2007 as follows: Let G = (V(G), E(G)) be a graph with vertex set V(G) and E(G) Given non-negative integers r,s and t, an [r,s,t]-colouring of a graph G = (V(G),E(G)) is a mapping C from V(G) ∪ E(G) to the colour set {0,1,2...,k − 1} such that |c(v _i ) − c(v _j )| ≥ r for every two adjacent vertices v _i , v _j , |c(e _i ) − c(e _j )| ≥ s for every two adjacent edges e _i , e _j , and |c(v _i ) − c(e _j )| ≥ t for all pairs of incident vertices and edges, respectively. The [r,s,t]-chromatic number X _r,s,t (G) of G is defined to be the mininum k such that G admits an [r,s,t]-colouring. In this paper, we determine the [r,s,t]-chromatic number for join graphs S _n + O _m