Preserving and Generalizing χ-boundedness

Abstract

The notion of χ-boundedness, introduced by Gyárfás in the mid-1980s, captures when, for every induced subgraph of a graph, large chromatic number can occur only due to the presence of a sufficiently large complete subgraph. The study of χ-boundedness is a central topic in graph theory. Understanding which hereditary classes of graphs are χ-bounded is of particular importance for advancing our understanding of how restrictions on the induced subgraphs of a graph affect both its global structure and key parameters such as the clique number and the independence number.

Which classes of graphs are χ-bounded? A method that has been used to prove that a class C of graphs is χ-bounded proceeds as follows: we prove that C can be obtained by applying operations that preserve χ-boundedness to already χ-bounded classes. This approach gives rise to the following question: which operations preserve χ-boundedness?

Given k graphs G₁,…,Gₖ, their intersection is the graph (∩{i∈[k]}V(Gᵢ), ∩{i∈[k]}E(Gᵢ)). Given k graph classes G₁,…,Gₖ, we call the class {G : ∀i∈[k], ∃Gᵢ∈Gᵢ such that G = G₁ ∩ ⋯ ∩ Gₖ} the graph-intersection of G₁,…,Gₖ. In the mid-1980s, in his seminal paper “Problems from the world surrounding perfect graphs”, Gyárfás observed that, due to early results of Asplund and Grünbaum, and Burling, graph-intersection does not preserve χ-boundedness in general, and he raised some questions regarding the interplay between graph-intersection and χ-boundedness. This topic has not received much attention since then. In this thesis, we formalize and explore the connection between the operation of graph-intersection and χ-boundedness.

Let r ≥ 2 be an integer. We denote by Kᵣ the complete graph on r vertices. The Kᵣ-free chromatic number of a graph G, denoted by χᵣ(G), is the minimum size of a partition of V(G) into sets each of which induces a Kᵣ-free graph. Generalizing χ-boundedness, we say that a class C of graphs is χᵣ-bounded if there exists a function f:ℕ→ℕ such that for every G∈C and every induced subgraph G′ of G, we have χᵣ(G′) ≤ f(ω(G′)), where ω(G′) denotes the clique number of G′. We study the induced subgraphs of graphs with large Kᵣ-free chromatic number.

Finally, we introduce the fractional Kᵣ-free chromatic number, and for every r ≥ 2 we construct K_{r+1}-free intersection graphs of straight-line segments in the plane with arbitrarily large fractional Kᵣ-free chromatic number.