The typical structure of dense claw-free graphs

TL;DR

The paper analyzes the enumeration and typical structure of dense claw-free graphs through graphon variational methods. It derives explicit entropy-density and large-deviation rate-formulas by solving constrained variational problems over claw-free graphons, revealing a second-order phase transition in edge-density and a first-order transition in claw-freeness probability. It shows that in the supercritical regime almost all claw-free graphs are co-bipartite with a unique optimal graphon, while in the subcritical regime the typical structure is a disjoint union of co-bipartite blocks and a sparse graph, both described by precise graphon limits. The work extends the graphon large-deviation framework to induced-subgraph constraints, providing sharp asymptotics, stability results, and a detailed structural picture for claw-free graphs with constant densities.

Abstract

We analyze the asymptotic number and typical structure of claw-free graphs at constant edge densities. The first of our main results is a formula for the asymptotics of the logarithm of the number of claw-free graphs of edge density $γ\in (0,1)$. We show that the problem exhibits a second-order phase transition at edge density $γ^\ast=\frac{5-\sqrt{5}}{4}$. The asymptotic formula arises by solving a variational problem over graphons. For $γ\geqγ^\ast$ there is a unique optimal graphon, while for $γ<γ^\ast$ there is an infinite set of optimal graphons. By analyzing more detailed structure, we prove that for $γ<γ^\ast$, there is in fact a unique graphon $W$ such that almost all claw-free graphs at edge density $γ$ are close in cut metric to $W$. We also analyze the probability of claw-freeness in the Erdős-Rényi random graph $G(n,p)$ for constant $p$, obtaining a formula for the large-deviation rate function for claw-freeness. In this case, the problem exhibits a first-order phase transition at $p^\ast=\frac{3-\sqrt{5}}{2}$, separating distinct structural regimes. At the critical point $p^\ast$, the corresponding graphon variational problem has infinitely many solutions, and we again pinpoint a unique optimal graphon that describes the typical structure of $G(n,p^\ast)$ conditioned on being claw-free.

Figures (3)

• Figure 1: The entropy density $r^\ast(\gamma)$ and rate function $r_\ast(p)$.
• Figure 2: Unique optimizers of the variational problem $\phi(\gamma)$ at edge densities $\gamma=\frac{5-\sqrt{5}}{4}$, $\frac{4}{5}$, $\frac{5}{6}$, and $\frac{13}{14}$. Each gray area represents a density in $[\frac{3-\sqrt{5}}{2},1]$ and black areas represent density 1.
• Figure 3: Examples of optimizers of the variational problem $\phi(\gamma)$ at edge density $\gamma=\frac{1}{4}$. The white, gray, and black areas represent densities $0$, $\frac{3-\sqrt{5}}{2}$, and $1$, respectively.

Theorems & Definitions (101)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Conjecture 1.9
• Theorem 2.1: Janson's inequality, riordan2015janson
• Lemma 2.2
• Definition 2.3: Type coloring and type of a graph