Backbone colouring conjecture for chordal graphs and spanning forests

Let $G$ be a chordal graph and let $H$ be a spanning forest of $G$. Here $`\operatorname{BBC}_2(G,H)` denotes the minimum number of colours in a colouring of$G$such that endpoints of every edge of$H$receive colours differing by at least$2$, and$\omega(G) is the clique number of $G$.

Backbone colouring conjecture.

$$\operatorname{BBC}_2(G,H)\leq \omega(G)+\mathcal O(1).$$

The paper presents this as an open conjecture motivated by the bound $\omega(G)+\mathcal O(\sqrt{\omega(G)})$ proved for spanning forests. It predicts a constant additive term independent of the chordal graph and its clique number.