On the perfect $k$-divisibility of graphs
David Scholz
TL;DR
This paper introduces a generalized notion of perfectly divisible graphs called perfectly $k$-divisible graphs, defined by a recursive condition on induced subgraphs and partitions into a smaller-clique part and a perfectly $(k-1)$-divisible part. It establishes a sharp χ-binding function for these graphs, proving that $\chi(G) \le \binom{\omega(G)+k-1}{k}$ via induction on the pair $(k, \omega)$, and provides a structural lemma showing how perfect $k$-divisibility propagates from $H$-free to $(H\cup K_1)$-free graphs. The work yields corollaries bounding $\chi(G)$ in terms of stability number $\alpha$, and connects to known χ-bounded classes ($2K_2$-free, $(P_n \cup P_2)$-free), while outlining several open questions and conjectures about related graph classes and divisibility properties.
Abstract
A graph $G$ is perfectly divisible if, for every induced subgraph $H$ of $G$, either $V(H)$ is a stable set or admits a partition into two sets $X_1$ and $X_2$ such that $ω(H[X_1]) < ω(H)$ and $H[X_2]$ is a perfect graph. In this article, we propose the following generalisation of perfectly divisible graphs. A graph $G$ is perfectly $1$-divisible if $G$ is perfect and perfectly $k$-divisible if, for every induced subgraph $H$ of $G$, either $V(H)$ is a stable set or admits a partition into two sets $X_1$ and $X_2$ such that $ω(H[X_1]) < ω(H)$ and $H[X_2]$ is perfectly $(k-1)$-divisible, $k \in \mathbb{N}_{> 1}$. Our main result establishes that every perfectly $k$-divisible graph $G$ satisfies $χ(G) \leq \binom{ω(G)+k-1}{k}$ which generalises the known bound for perfectly divisible graphs.