4-colorability of (P7,C3)-free graphs

Determine whether every (P_7,C_3)-free graph has chromatic number at most 4, extending the known upper bound of 5 for this graph class.

Background

The paper establishes that every (P_2\cup P_4,C_3)-free graph has chromatic number at most 4 and observes that the graph classes considered are subclasses of the broader class of (P_7,C_3)-free graphs. The cited known result gives only the upper bound \chi(G)\leq 5 for the latter class, leaving open whether the stronger bound 4 holds universally.

References

An interesting problem is that whether every $(P_7,C_3)$-free graph $G$ satisfies $\chi(G)\leq4$?

— Structure, Perfect Divisibility and Coloring of ($P_2\cup P_4, C_3$)-Free Graphs  (2509.14135 - Chen et al., 17 Sep 2025) in Remark following the proof of Theorem 2, concluding section