Backbone coloring of graphs with galaxy backbones

Abstract

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 u v ∈ 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 ) | ( q ≤ | c ( u ) − c ( v ) | ≤ k − q ), for every edge u v ∈ E ( H ) . The (circular) q -backbone chromatic number of ( G , H ) , denoted by BBC q ( G , H ) ( CBC q ( 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: CBC q ( G , H ) ≤ 2 q + 2 when q ≥ 3 and H is a galaxy; CBC q ( G , H ) ≤ 2 q when q ≥ 4 and H is a matching; and CBC 3 ( 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 BBC 2 ( G , M ) ≤ Δ ( G ) + 1 , for any matching M of an arbitrary graph G (Miškuf et al., 2010).