[r,s,t,]-coloring Of Some Graphs

Let G=(V,E) be a simple graph with vertex set V and edge set E. Given non-negative integers r, s and t, an [r,s,t]-coloring of a graph G is a mapping c from V(G)∪V(E) to the color set {0,1,K,k-1} such that|c(vi)-c(vj)|≥r for every two adjacent vertices vi, vj∈V,|c(ei)-c(ej)|≥s for every adjacent edges ei, ej∈E, and c(vi)-c(ej)|≥t for every pair of incident vertex and edge, respectively. From the definition, we see that [r,s,t]-coloring is a generalization of the total coloring and the classical vertex and edge colorings of graphs. The [r,s,t]-chromatic number Xr,s,t(G) of G is defined to be the minimum k such that G admits an [r,s,t]-coloring. In this thesis, we discuss [r,s,t]-coloring of stars and complete bipartite graphs.