Chromatic bound for (P7, K4-e)-free graphs

Determine whether every (P7, K4-e)-free graph G satisfies chi(G) <= omega(G) + epsilon for a positive integer epsilon >= 3.

Background

The paper observes that every (P2 + P4, K4-e)-free graph is also (P7, K4-e)-free. Its main theorem therefore provides a partial answer to an unpublished question attributed to Ingo Schiermeyer concerning whether the chromatic number of all (P7, K4-e)-free graphs can be bounded by their clique number plus a fixed additive constant.

The result established in the paper does not settle the full (P7, K4-e)-free class, so the proposed universal chromatic bound remains unresolved.

References

Is it true that, every (P_7, K_4-e)-free graph G satisfies $\chi(G) \leq \omega(G) + \varepsilon$, where $\varepsilon\geq 3$ is a positive integer?

— ($P_2+P_4$, $K_4-e$)-free graphs are nearly $ω$-colorable  (2501.02543 - Angeliya et al., 5 Jan 2025) in Problem 2, Introduction