Near-optimal colorability of (F, K_l-e)-free graphs

Determine whether, for every forest F that is not an induced subgraph of P4 and every integer l >= 4, there exists a constant c such that every (F, K_l-e)-free graph G satisfies chi(G) <= max{c, omega(G)}, and, if such a constant exists, determine the smallest possible c.

Background

The paper discusses the notion of near-optimal colorability, in which the chromatic number of every graph in a class is bounded by the maximum of a fixed constant and the graph's clique number. Ju and Huang posed the general question for classes of (F, K_l-e)-free graphs, where F is a forest not induced by P4 and l >= 4.

The paper notes that the status of this question is unknown even for the specific case F = P2 + P4 and l = 4. The authors partially resolve that case by proving chi(G) <= max{6, omega(G)} for (P2 + P4, K4-e)-free graphs with omega(G) >= 3, but the broader formulation and the optimal constant remain unresolved.

References

Let G be the class of (F, K_{\ell}-e)-free graphs, where F is a forest which is not an induced subgraph of a P_4 and \ell\geq 4. Does there exist a constant c\in \mathbb{N} such that every G\in \mathcal G satisfies \chi(G) \leq \max{c,~\omega(G)}? If so, what is the smallest possible constant c?

— ($P_2+P_4$, $K_4-e$)-free graphs are nearly $ω$-colorable  (2501.02543 - Angeliya et al., 5 Jan 2025) in Problem 1, Introduction (labeled JH-Problem)