Intersections of graphs and $χ$-boundedness

TL;DR

The paper investigates how the graph-intersection operation interacts with $χ$-boundedness, introducing intersectionwise $χ$-guarding and intersectionwise self-$χ$-guarding classes. It develops a decomposition-based framework showing decomposable graph classes are intersectionwise $χ$-guarding, and identifies several χ-bounded families that inherit this property (e.g., unit interval graphs and line graphs of bipartite graphs), while giving negative results via Burling-type constructions for other classes. A complete characterization is provided for single forbidden subgraph families: a class of $H$-free graphs is intersectionwise $χ$-guarding if and only if $H$ is a forest on at most three vertices ($P_{2}$, $P_{3}$) or a disjoint union of isolated vertices ($rK_{1}$). The paper also studies intersectionwise self-$χ$-guarding classes, proposing constructions to generate new such classes from existing ones and proving several structural results for $K_{1,t}$-free, $P_{4}$-free, and complete multipartite families, with potential implications for the broader χ-boundedness program.

Abstract

Given $k$ graphs $G_{1}, \ldots, G_{k}$, their intersection is the graph $(\cap_{i\in [k]}V(G_{i}), \cap_{i\in [k]}E(G_{i}))$. Given $k$ graph classes $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$, we call the class $\{G: \forall i \in[k], \exists G_{i} \in \mathcal{G}_{i} \text{ such that } G=G_{1}\cap \ldots \cap G_{k}\}$ the graph-intersection of $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$. The main motivation for the work presented in this paper is to try to understand under which conditions graph-intersection preserves $χ$-boundedness. We consider the following two questions: Which graph classes have the property that their graph-intersection with every $χ$-bounded class of graphs is $χ$-bounded? We call such a class intersectionwise $χ$-guarding. We prove that classes of graphs which admit a certain kind of decomposition are intersectionwise $χ$-guarding. We provide necessary conditions that a finite set of graphs $\mathcal{H}$ should satisfy if the class of $\mathcal{H}$-free graphs is intersectionwise $χ$-guarding, and we characterize the intersectionwise $χ$-guarding classes which are defined by a single forbidden induced subgraph. Which graph classes have the property that, for every positive integer $k$, their $k$-fold graph-intersection is $χ$-bounded? We call such a class intersectionwise self-$χ$-guarding. We study intersectionwise self-$χ$-guarding classes which are defined by a single forbidden induced subgraph, and we prove a result which allows us construct intersectionwise self-$χ$-guarding classes from known intersectionwise $χ$-guarding classes.

Figures (2)

• Figure 1: From left to right: The net, the complement of the graph $H$, and the complement of the graph $X_{96}$. Each of these graphs is a strongly chordal graph which is not a Burling strongly chordal graph.
• Figure 2: An illustration of the situation in the proof of \ref{['cl:good.pair']}. By the definition of $\tilde{B_{l'}}$ we have that $v$ is anticomplete to $\tilde{B_{l'}}$. Thus both $v$ and $x$ are anticomplete to $\tilde{B_{l'}}\setminus N_{G_{p}}(x)$. The disjoint union of an induced $H_{j}+(r_{j}^{p}-1)K_{2}$ in $\tilde{B_{l'}}\setminus N_{G_{p}}(x)$ with the edge $vx$ would result in an induced $H_{j}+r_{j}^{p}K_{2}$ in $G_{p}$.

Theorems & Definitions (105)

• Proposition 1.1: Folklore
• Proposition 1.2: Gyárfás gyarfas1985problems
• Theorem 2.1
• proof : Proof of \ref{['obs:chig.is.chib']}
• proof : Proof of \ref{['thm:chi_imposing.characterization']}
• Theorem 3.1
• Lemma 3.2
• Proposition 3.3
• proof : Proof of \ref{['prop:bound.clique.of.H']}
• Theorem 3.4: Ramsey ramsey1930problem