Towards Esperet's Conjecture: Polynomial $\chi$-Bounds for Structured Graph Classes

TL;DR

This paper proves polynomial and linear bounds on the chromatic number for certain structured graph classes, extending Esperet's conjecture and broadening understanding of graph coloring constraints in hereditary classes.

Contribution

It establishes new polynomial and linear chromatic bounds for classes of graphs defined by forbidden subgraphs and structural properties, extending previous results on broom-free graphs.

Findings

Polynomial $oldsymbol{ ext{chi}}$-bounds for $oldsymbol{ ext{B}^{+}(p+2, t-1)}$-free graphs

Linear $oldsymbol{ ext{chi}}$-bounds for $K_3(t)$-free graphs

Extension of bounds to hereditary classes with specific forbidden subgraphs

Abstract

In this paper, we establish that the class of $\{P_6, (2,2)\text{-broom}\}$-free graphs contains a subclass $\mathcal{L}_i$, defined by certain cutset conditions, whose chromatic number admits a linear $\chi$-bound. Building on recent results showing that broom-free graphs excluding $K_d(t)$ as a subgraph admit a polynomial bound in~$t$ on their chromatic number (A broom is obtained from a path with one end $v$ by adding leaves adjacent to $v$), we extend this result to the hereditary class $\mathcal{H}$ of $C_4$-free and \emph{$p$-flag}-free graphs (where a \emph{$p$-flag} is a triangle with an attached $p$-path). We show that if $G \in \mathcal{H}$ is $B^{+}(p+2, t-1)$-free (for $p \ge 2$ and $t \ge 3$, that is, if it excludes a generalized broom with an additional leaf), and does not contain $K_d(t)$ as a subgraph, then $\chi(G)$ is polynomially bounded in $t$. Furthermore, for the subclass of $\mathcal{H}$ excluding $K_3(t)$ as a subgraph, we prove that $\chi(G)$ is linearly $\chi$-bounded in $\omega(G)$.