Steinberg-like theorems for backbone colouring

Abstract

A function f:V(G)→{1,…,k} is a (proper) k-colouring of G if ∣f(u)−f(v)∣≥1, for every edge uv∈E(G). The chromatic numberχ(G) is the smallest integer k for which there exists a proper k-colouring of G.Given a graph G and a subgraph H of G, a circular q-backbone k-colouring f of (G,H) is a k-colouring of G such that q≤∣f(u)−f(v)∣≤k−q, for each edge uv∈E(H). The circularq-backbone chromatic number of a graph pair (G,H), denoted CBCq(G,H), is the minimum k such that (G,H) admits a circular q-backbone k-colouring.Inspired by Steinberg’s conjecture, we conjecture that if G is a planar graph containing no cycles on 4 or 5 vertices and H⊆G is a forest, then CBC2(G,H)≤6. In this work, we first show that if G is a planar graph containing no cycle on 4 or 5 vertices and H⊆G is a forest, then CBC2(G,H)≤7. Then, we prove that if H⊆G is a forest whose connected components are paths, then CBC2(G,H)≤6.