Characterization of 4-chromatic (P7,C3)-free graphs

Characterize the graphs with chromatic number 4 among (P_7,C_3)-free graphs, beyond the Mycielski-Grötzsch graph, conditional on the conjectured 4-colorability of the entire class.

Background

The paper notes that the Mycielski-Grötzsch graph is a 4-vertex-critical example in the narrower class of (P_2\cup P_4,C_3)-free graphs and that the broader class of (P_7,C_3)-free graphs is known to be 5-colorable. If every (P_7,C_3)-free graph were shown to be 4-colorable, a subsequent unresolved problem would be to determine all 4-chromatic graphs in that class and identify examples other than the Mycielski-Grötzsch graph.

References

If the answer is yes, then a further problem is that which graphs have chromatic number 4 other than Mycielski-Grötzsch graph.

— 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