Pollyanna and Polynomially \c{hi}-Bounded Graph Classes

TL;DR

This work advances the Pollyanna framework by deriving polynomial $χ$-boundedness and strengthened Pollyanna results for several hereditary graph classes defined by forbidden induced subgraphs. The authors introduce a clique-centered decomposition around a maximum clique and a suite of structural templates (including $t$-lolli­pops, $F(3,l)$, and hammer$(t)^+$) along with a robust $P$-property toolkit to bound chromatic number via the clique number $ω$. They prove, in particular, that $ ext{Diamond}$-free graphs forbidding hammer$(t)^+$ are $t$-strongly Pollyanna, that certain bowtie/dumbbell exclusions yield $(2t-2)$-strongly Pollyanna, and that several specific forbidden-structure classes are polynomially $χ$-bounded. Additionally, a diamond-free, two-triangle-per-edge setting with extra forbidden configurations yields polynomial $χ$-boundedness, highlighting how local structure enforces global colorability. These results deepen the internal theory linking $χ$-boundedness, the Pollyanna property, and explicit forbidden-subgraph characterizations.

Abstract

A hereditary graph class is called polynomially $χ$-bounded if there exists a polynomial function $f$ such that $χ(G) \le f(ω(G))$ for every induced subgraph $G$. A class $\mathcal{C}$ is called Pollyanna if, for every $χ$-bounded class $\mathcal{F}$, the class $\mathcal{C} \cap \mathcal{F}$ is polynomially $χ$-bounded. In the paper by Chudnovsky et al., \emph{Reuniting $χ$-boundedness with polynomial $χ$-boundedness} (J.\ Combin.\ Theory Ser.\ B 176 (2026), 30--73), the authors posed twelve problems and one conjecture concerning the Pollyanna framework. In this work, we investigate several of these problems by studying the chromatic number of hereditary graph classes defined by forbidden induced subgraphs. We prove three new strong Pollyanna results. In particular, for every $t \ge 2$, every $\{\text{diamond}, \mathrm{hammer}(t)^+\}$-free graph is $t$-strongly Pollyanna. We also show that graph classes obtained by forbidding suitable combinations of bowties and dumbbells are $(2t-2)$-strongly Pollyanna. We show that the class of $\{(2,2)$-bowtie, $P_5$, $(3,3)$-dumbbell$\}$-free graphs is polynomially $χ$-bounded. We also prove polynomial $χ$-boundedness for diamond-free graphs in which every edge lies in at least two triangles, under additional forbidden configurations.

Figures (16)

• Figure 1.1:
• Figure 2.1:
• Figure 2.3:
• Figure 2.4: $P3$ (for $(3,t)$-lollipop)
• Figure 2.6: $P5$

Theorems & Definitions (19)

• Definition 1.1
• Theorem 1.2: Chudnovsky2026reuniting
• Proposition 1.3: Erdos1935
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• proof