Perfect Divisibility and Coloring of Some Bull-Free Graphs

TL;DR

This work extends the theory of perfect divisibility to broader bull-free graph families by introducing and exploiting the related notion of perfect weight divisibility. It proves that $(\text{odd torch}, bull)$-free graphs and $(P_8,C_5,bull)$-free graphs are perfectly divisible, and that $(P_6,bull)$-free graphs are perfectly divisible iff they are $F$-free, with corollaries giving strong coloring bounds such as $\chi(G)\le \binom{\omega(G)+1}{2}$ and $\chi(G)\le \omega(G)^7$ for relevant classes. The approach hinges on structural analysis with locally perfect neighborhoods, homogeneous sets, and avoidance of certain induced subgraphs, combined with weight-based divisibility arguments to rule out minimally non-divisible configurations. These results generalize existing bull-free coloring bounds and provide a unified framework yielding practical color bounds in several graph families. The work also highlights the role of the Mycielski-Grötzsch graph as a barrier to perfect divisibility in $(P_6,bull)$-free graphs and demonstrates how substitution-closure interacts with $\,\chi$-boundedness.

Abstract

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 $ω(H[B])<ω(H)$. A {\em bull} is a graph consisting of a triangle with two disjoint pendant edges, a {\em fork } is a graph obtained from $K_{1,3}$ by subdividing an edge once, and an {\em odd torch} is a graph obtained from an odd hole by adding an edge $xy$ such that $x$ is non-adjacent to any vertex on the odd hole and the set of neighbors of $y$ on the odd hole is a stable set. Chudnovsky and Sivaraman [J. Graph Theory 90 (2019) 54-60] proved that every (odd hole, bull)-free graph and every ($P_5$, bull)-free graph are perfectly divisible. Karthick {\em et al.} [The Electron. J. of Combin. 29 (2022) P3.19.] proved that every (fork, bull)-free graph is perfectly divisible. Chen and Xu [Discrete Appl. Math. 372 (2025) 298-307.] proved that every ($P_7,C_5$, bull)-free graph is perfectly divisible. Let $H\in$\{\{odd~torch\}, $\{P_8,C_5\}\}$. In this paper, we prove that every ($H$, bull)-free graph is perfectly divisible. We also prove that a ($P_6$, bull)-free graph is perfectly divisible if and only if it contains no Mycielski-Grötzsch graph as an induced subgraph. As corollaries, these graphs are $\binom{ω+1}{2}$-colorable. Notice that every odd torch contains an odd hole, an induced $P_5$, and an induced fork. Therefore, our results generalize their findings. Moreover, we prove that every ($P_6$, bull)-free graph $G$ satisfies $χ(G)\leqω(G)^7$.

Figures (1)

• Figure 1: Illustration of some graphs.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.1
• Corollary 1.2
• Theorem 1.5
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3