[Paper Review] Graph limits and hereditary properties

Svante Janson|arXiv (Cornell University)|Feb 17, 2011

Advanced Graph Theory Research24 references4 citations

TL;DR

This paper investigates graph limits associated with hereditary graph classes, focusing on structural constraints such as forbidden subgraphs and intersection graph representations. It establishes that for certain hereditary classes—like claw-free and coclaw-free graphs—graph limits converge to either the empty graph or complete graph limit, with random limits emerging when both a graph and its complement are in the class.

ABSTRACT

We collect some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs, chordal graphs, cographs, interval graphs, unit interval graphs, threshold graphs, and line graphs.

Motivation & Objective

• To characterize graph limits of hereditary graph classes using the framework of graph limit theory.
• To analyze how structural constraints—such as forbidden subgraphs or intersection properties—affect the limiting behavior of sequences of graphs.
• To identify conditions under which graph limits are deterministic (e.g., [0] or [1]) versus random (e.g., mixture of [0] and [1]).
• To extend known results on specific graph classes (e.g., interval graphs, cographs) to a general framework of hereditary properties.
• To establish sharp bounds on edge counts in graphs that are both claw-free and coclaw-free, linking extremal graph theory with limit theory.

Proposed method

• Uses the compact metric space of graph limits, $υ\overline{\mathcal{U}}$, with convergence defined via homomorphism densities $t(F,G)$ and induced subgraph densities $t_{\text{ind}}(F,G)$.
• Applies the theory of graphons to represent graph limits, where a graphon $W$ is a symmetric measurable function $[0,1]^2 \to [0,1]$.
• Characterizes limits of hereditary classes via random graph models, showing that limits are determined by the asymptotic behavior of $t(F,G_n)$ as $n \to \infty$.
• Employs extremal graph theory results, particularly Ramsey theory, to bound the structure of graphs that are both claw-free and coclaw-free.
• Uses symmetry and duality between a graph and its complement to show that limits of graphs closed under complementation can be random mixtures of [0] and [1].
• Applies measure-theoretic arguments to show that if a graphon $W$ is both claw-free and coclaw-free, then $W$ must be 0 or 1 almost everywhere, leading to trivial limits.

Experimental results

Research questions

• RQ1 Under what conditions do sequences of graphs in a hereditary class converge to a unique graph limit?
• RQ2 How do structural constraints like being claw-free or coclaw-free affect the possible graph limits of a hereditary class?
• RQ3 What is the limiting behavior of uniformly random graphs in a hereditary class that is closed under taking complements?
• RQ4 Can the edge density of graphs that are both claw-free and coclaw-free be bounded in a way that forces convergence to a trivial limit?
• RQ5 To what extent do intersection graph representations (e.g., interval, unit interval) yield non-trivial graph limits?

Key findings

• A graph limit that is both claw-free and coclaw-free must be almost everywhere 0 or 1, hence the only possible limits are [0] (empty) or [1] (complete).
• For any sequence of graphs that are both claw-free and coclaw-free, the number of edges satisfies $e(G_n) = o(n^2)$ or $inom{n}{2} - e(G_n) = o(n^2)$, implying convergence to [0] or [1].
• If $n \geq R(8,8)$, then a graph $G$ is both claw-free and coclaw-free if and only if $G \in \mathcal{Q}$ or $\overline{G} \in \mathcal{Q}$, where $\mathcal{Q}$ is the class of graphs with maximum degree ≤2 and no $K_3$ components.
• The only graph limit in the closure of $\mathcal{Q}$ is $[0]$, so any sequence in $\mathcal{Q}$ converges to the empty graph limit.
• For uniformly random claw-free and coclaw-free graphs $G_n$, the limit in distribution is a random graph limit that equals [0] or [1] with equal probability $1/2$.
• The result that $\widehat{\mathcal{Q}} = \{[0]\}$ implies that no non-trivial graph limit arises from the class of graphs with components restricted to paths and cycles of length ≥4.