Backbone coloring of graphs with galaxy backbones

A (proper) k-coloring of a graph G = (V, E) is a function c : V (G) -> {1, ... , k} such that c(u) not equal c(v), for every uv is an element of 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 <= vertical bar c(u) - c(v)vertical bar (q <= vertical bar c(u) - c(v)vertical bar <= k-q), for every edge uv is an element of E(H). The (circular) q-backbone chromatic number of (G, H), denoted by BBCq(G, H) (CBCq(G, H)), is the minimum integer k for which there exists a (circular) q-backbone k-coloring of (G, H). In this work, we (partially) answer three questions posed by Havet et al. (2014), namely, we prove that if G is a planar graph, H is a spanning subgraph of G and q is a positive integer, then: CBCq(G, H) <= 2q + 2 when q >= 3 and H is a galaxy; CBCq(G, H) <= 2q when q >= 4 and H is a matching; and CBC3(G, H) <= 7 when H is a matching and G has no triangles sharing an edge. In addition, we present a polynomial-time algorithm to determine both parameters for any pair (G, H), whenever G has bounded treewidth. Finally, we show how to fix a mistake in a proof that BBC2(G, M) <= Delta(G) + 1, for any matching M of an arbitrary graph G (Miskuf et al., 2010). (c) 2022 Elsevier B.V. All rights reserved.