On (d,1)-total numbers of graphs

AbstractA (d,1)-total labelling of a graph G assigns integers to the vertices and edges of G such that adjacent vertices receive distinct labels, adjacent edges receive distinct labels, and a vertex and its incident edges receive labels that differ in absolute value by at least d. The span of a (d,1)-total labelling is the maximum difference between two labels. The (d,1)-total number, denoted λdT(G), is defined to be the least span among all (d,1)-total labellings of G. We prove new upper bounds for λdT(G), compute some λdT(Km,n) for complete bipartite graphs Km,n, and completely determine all λdT(Km,n) for d=1,2,3. We also propose a conjecture on an upper bound for λdT(G) in terms of the chromatic number and the chromatic index of G

[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