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

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