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

Abstract

Goedgebeur and Schaudt [J. Graph Theory 87 (2018) 188-207] conjectured that all 4-vertex-critical $(P_7,C_3)$-free graphs belongs to the family $\cal G$, which consists of seven explicitly defined graphs. In this paper, we establish a structural decomposition for $(P_2\cup P_4,C_3)$-free graphs and show that the conjecture holds for this class. Consequently, we determine the chromatic number of $(P_2\cup P_4, C_3)$-free graphs. A graph $G$ is {\em perfectly divisible} if for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. A {\em bull} is a graph consisting of a triangle with two disjoint pendant edges. Notice that the class of $(P_2\cup P_4, C_3)$-free graphs is a subclass of ($P_2\cup P_4$, bull)-free graphs. In this paper, we prove that a ($P_2\cup P_4$, bull)-free graph is perfectly divisible if and only if it contains no Mycielski-Gr\"{o}tzsch graph. This generalizes the main result of Deng and Chang [Graphs Combin. (2025) 41: 63].