The λ-backbone colorings of graphs with tree backbones

The λ-backbone coloring is one of the various problems of vertex colorings in graphs. Given an integer λ ≥ 2, a graph G = (V,E), and a spanning subgraph (backbone) H = (V, EH) of G, a λ-backbone coloring of (G,H) is a proper vertex coloring V → {1, 2, ...} of G in which the colors assigned to adjacent vertices in H differ by at least λ. The λ-backbone coloring number BBCλ(G,H) of (G,H) is the smallest positive integer l for which there exists a λ-backbone coloring f : V → {1, 2, ..., l} of (G,H). For a graph G with chromatic number χ(G) = k, the λ-backbone coloring number is denoted by BBCλ(G(k),H). In this paper, we consider λ-backbone colorings of graphs with tree backbones. We determine the relation between χ(G) and BBCλ(G(k),H) of (G,H) with a tree backbone H for λ ≥ 3.