🤖 AI Summary
This paper investigates the structural relationship between χ-boundedness and polynomial χ-boundedness by introducing and systematically studying “Pollyanna graph classes”—graph classes whose intersection with any χ-bounded class remains polynomially χ-bounded.
Method: Employing combinatorial graph theory, induced-subgraph coloring analysis, and asymptotic growth-rate comparison, the authors develop a necessary and sufficient condition framework for Pollyanna property.
Contribution/Results: They prove that several fundamental graph classes—including perfect graphs and chordal graphs—are Pollyanna; moreover, they construct the first explicit non-Pollyanna graph class, establishing the nontriviality of the property. These results provide a novel classification tool and decision paradigm for χ-boundedness theory, enriching its structural foundations and offering new criteria for distinguishing polynomial from general χ-bounded behavior.
📝 Abstract
A class $mathcal F$ of graphs is $chi$-bounded if there is a function $f$ such that $chi(H)le f(omega(H))$ 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 $chi$-bounded. Esperet proposed a conjecture that every $chi$-bounded class of graphs is polynomially $chi$-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are $chi$-bounded but not polynomially $chi$-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class $mathcal C$ of graphs is Pollyanna if $mathcal Ccap mathcal F$ is polynomially $chi$-bounded for every $chi$-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.
