Structure, Coloring, and Perfect Divisibility of $(P_2\cup P_4, C_3)$-Free Graphs

Abstract

Goedgebeur and Schaudt [J. Graph Theory 87 (2018), 188-207] conjectured that every $4$-vertex-critical $(P_7,C_3)$-free graph belongs to a family of seven explicitly defined graphs. In this paper, we establish a structural theorem for connected $(P_2\cup P_4,C_3)$-free graphs. As a consequence, we prove that the Mycielski-Grötzsch graph is the unique $4$-vertex-critical graph in this class, thereby confirming the conjecture of Goedgebeur and Schaudt for $(P_2\cup P_4,C_3)$-free graphs. Our structural theorem also yields a characterization of the chromatic number of these graphs and an $O(n^4)$-time algorithm for deciding whether an $n$-vertex $(P_2\cup P_4,C_3)$-free graph is $3$-colorable. We further study perfect divisibility in the larger class of $(P_2\cup P_4,\text{bull})$-free graphs. We prove that a $(P_2\cup P_4,\text{bull})$-free graph is perfectly divisible if and only if it is Mycielski-Grötzsch graph-free. This result generalizes the main theorem of Deng and Chang [Graphs Combin. 41 (2025), 63].