Reuniting $χ$-boundedness with polynomial $χ$-boundedness

TL;DR

This work investigates the boundary between χ-boundedness and polynomial χ-boundedness by introducing Pollyanna classes: graph classes C such that C ∩ F is polynomially χ-bounded for every χ-bounded class F. It proves that several natural hereditary classes are Pollyanna with explicit n-strong bounds, including mK_t-free, (t,k)-pineapple-free, t-lollipop-free, bowtie-free, and bull-free graphs, by leveraging structural decompositions, substitutions, and trigraph techniques. It also establishes non-Pollyanna phenomena: for certain finite families not containing any willow, the F-free graphs fail to be Pollyanna, highlighting the sharp contrast between global and local χ-boundedness and revealing the central role of willows in this theory. The results provide a unified framework connecting χ-boundedness, polynomial bounds, and graph decompositions, with implications for understanding Erdős–Hajnal-type phenomena and related coloring questions in restricted graph classes.

Abstract

A class $\mathcal{F}$ of graphs is $χ$-bounded if there is a function $f$ such that $χ(H)\le f(ω(H))$ for all induced subgraphs $H$ of a graph in $\mathcal{F}$. If $f$ can be chosen to be a polynomial, we say that $\mathcal{F}$ is polynomially $χ$-bounded. Esperet proposed a conjecture that every $χ$-bounded class of graphs is polynomially $χ$-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are $χ$-bounded but not polynomially $χ$-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class $\mathcal{C}$ of graphs is Pollyanna if $\mathcal{C}\cap \mathcal{F}$ is polynomially $χ$-bounded for every $χ$-bounded class $\mathcal{F}$ of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna.

Figures (12)

• Figure 1: Forbidding any of these graphs makes a Pollyanna class of graphs.
• Figure 2: A pentagram spider, a tall strider, and a short strider are graphs obtained from the above figure by adding any additional edges between two red hollow vertices.
• Figure 3: Graphs $\overline{P_9}$, $\overline{C_7}$, $F_7$, and $W_7$. The class of ($\overline{P_9}$, $\overline{C_7}$, $F_7$, $W_7$)-free graphs is not Pollyanna.
• Figure 4: An illustration for the proof of \ref{['prop:pineapple']}.
• Figure 5: A homogeneous pair.

Theorems & Definitions (125)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: Chudnovsky, Penev, Scott, and Trotignon CPST2013
• Theorem 2.2: Seinsche Seinsche1974
• Theorem 2.3: Chvátal and Sbihi chvatal1987bull
• Lemma 2.4: Lovász Lovasz1972
• Proposition 3.1
• proof
• Proposition 3.2