Steinberg-like theorems for backbone colouring

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 number χ(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 c of (G, H) is a k-colouring of G such that q≤|c(u)−c(v)|≤k−q, for each edge uv∈E(H). The circular q-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.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.