[r,s,t][r,s,t]-coloring of trees and bipartite graphs

Let G=(V,E)G=(V,E) be a graph with vertex set VV and edge set EE. Given non-negative integers rr, ss and tt, an [r,s,t][r,s,t]-coloring   of a graph GG is a function cc from V(G)∪E(G)V(G)∪E(G) to the color set {0,1,…,k−1}{0,1,…,k−1} such that |c(vi)−c(vj)|≥r|c(vi)−c(vj)|≥r for every two adjacent vertices vivi, vj∈Vvj∈V, |c(ei)−c(ej)|≥s|c(ei)−c(ej)|≥s for every two adjacent edges eiei, ej∈Eej∈E, and |c(vi)−c(ej)|≥t|c(vi)−c(ej)|≥t for every vertex vivi and its incident edge ejej. Thus, an [r,s,t][r,s,t]-coloring is a generalization of the total coloring and the classical vertex and edge colorings of graphs. The [r,s,t][r,s,t]-coloring can have many applications in different fields like scheduling [A. Kemnitz, M. Marangio, [r,s,t][r,s,t]-colorings of graphs, Discrete Mathematics 307 (2) (2007) 199–207], channel assignment problem [F. Bazzaro, M. Montassier, A. Raspaud, (d,1)(d,1)-total labelling of planar graphs with large girth and high maximum degree, Discrete Mathematics 307 (2007) 2141–2151], etc. The [r,s,t][r,s,t]-chromatic number  χr,s,t(G)χr,s,t(G) of GG is the minimum kk such that GG admits an [r,s,t][r,s,t]-coloring. In our paper, we give exact values (or bounds in one case) of the [r,s,t][r,s,t]-chromatic number of stars, for every positive rr, ss and tt. We also provide exact values and some tight bounds of this parameter for trees and bipartite graphs.