Backbone colouring: Tree backbones with small diameter in planar graphs

Given a graph $G$ and a spanning subgraph $T$ of $G$, a backbone $k$-coloring for $(G,T)$ is a mapping $c:V(G)\to\{1,\ldots,k\}$ such that $|c(u)-c(v)|\geq 2$ for every edge $uv\in E(T)$ and $|c(u)-c(v)|\geq 1$ for every edge $uv\in E(G)\setminus E(T)$. The backbone chromatic number $\chi_{bb}(G,T)$ is the smallest integer $k$ such that there exists a backbone $k$-coloring of $(G,T)$. In 2007, Broersma et al. \cite{BFG+07} conjectured that $\chi_{bb}(G,T)\leq 6$ 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.