Circular Backbone Colorings: on matching and tree backbones of planar graphs.

A (proper) k-coloring of a graph G=(V,E) is a function c:V(G)→{1,…,k} such that c(u)≠c(v) for every uv∈E(G). Given a graph G and a spanning subgraph H of G, a circular q-backbone k-coloring of (G,H) is a k-coloring c of G such that q≤|c(u)−c(v)|≤k−q for every edge uv∈E(H). The circular q-backbone chromatic number of (G,H), denoted by CBCq(G,H), is the minimum integer k for which there exists a circular q-backbonek-coloring of (G,H).