Backbone colorings of graphs with bounded degree

Abstract

We study backbone colorings, a variation on classical vertex colorings: Given a graph G and a subgraph H of G (the backbone of G ), a backbone coloring for G and H is a proper vertex k -coloring of G in which the colors assigned to adjacent vertices in H differ by at least 2 . The minimal k ∈ N for which such a coloring exists is called the backbone chromatic number of G . We show that for a graph G of maximum degree Δ where the backbone graph is a d -degenerated subgraph of G , the backbone chromatic number is at most Δ + d + 1 and moreover, in the case when the backbone graph being a matching we prove that the backbone chromatic number is at most Δ + 1 . We also present examples where these bounds are attained. Finally, the asymptotic behavior of the backbone chromatic number is studied regarding the degrees of G and H . We prove for any sparse graph G that if the maximum degree of a backbone graph is small compared to the maximum degree of G , then the backbone chromatic number is at most Δ ( G ) − Δ ( G ) .