[r,s,t][r,s,t]-Chromatic numbers and hereditary properties of graphs

Given non-negative integers r, s, and t  , an [r,s,t][r,s,t]-coloring   of a graph G=(V(G),E(G))G=(V(G),E(G)) is a mapping c   from V(G)∪E(G)V(G)∪E(G) to the color set {0,1,…,k-1}{0,1,…,k-1}, k∈Nk∈N, such that |c(vi)-c(vj)|⩾r|c(vi)-c(vj)|⩾r for every two adjacent vertices vi,vjvi,vj, |c(ei)-c(ej)|⩾s|c(ei)-c(ej)|⩾s for every two adjacent edges ei,ejei,ej, and |c(vi)-c(ej)|⩾t|c(vi)-c(ej)|⩾t for all pairs of incident vertices and edges, respectively. The [r,s,t][r,s,t]-chromatic number  χr,s,t(G)χr,s,t(G) of G is defined to be the minimum k such that G   admits an [r,s,t][r,s,t]-coloring.We characterize the properties O(r,s,t,k)={G:χr,s,t(G)⩽k}O(r,s,t,k)={G:χr,s,t(G)⩽k} for k=1,2,3k=1,2,3 as well as for k⩾3k⩾3 and max{r,s,t}=1max{r,s,t}=1 using well-known hereditary properties. The main results for k⩾3k⩾3 are summarized in a diagram.