Backbone colouring: Tree backbones with small diameter in planar graphs

Given a graph G and a spanning subgraph T of G, a backbone k-colouring for (G,T) is a mapping c:V(G)→{1,…,k} such that |c(u)−c(v)|≥2 for every edge uv∈E(T) and |c(u)−c(v)|≥1 for every edge uv∈E(G)∖E(T). The backbone chromatic number BBC(G,T) is the smallest integer k such that there exists a backbone k-colouring of (G,T). In 2007, Broersma et al. [2] conjectured that BBC(G,T)≤6 for every planar graph G and every spanning tree T of G. In this paper, we prove this conjecture when T has diameter at most four.