Graph limits of random graphs from a subset of connected $k$-trees

TL;DR

This paper studies the limiting behavior of a class of random connected $k$-trees with constraints on clique counts, showing convergence to the Continuum Random Tree and establishing local limits.

Contribution

It introduces a new model of constrained random $k$-trees and proves their global and local convergence to well-known infinite structures.

Findings

Scaled random $ ext{Omega}$-$k$-trees converge to the Continuum Random Tree.

Rooted $ ext{Omega}$-$k$-trees exhibit local convergence to an infinite random $ ext{Omega}$-$k$-tree.

Results extend understanding of the asymptotic structure of constrained random graphs.

Abstract

For any set $\Omega$ of non-negative integers such that $\{0,1\}\subseteq \Omega$ and $\{0,1\}\ne \Omega$, we consider a random $\Omega$-$k$-tree ${\sf G}_{n,k}$ that is uniformly selected from all connected $k$-trees of $(n+k)$ vertices where the number of $(k+1)$-cliques that contain any fixed $k$-clique belongs to $\Omega$. We prove that ${\sf G}_{n,k}$, scaled by $(kH_{k}\sigma_{\Omega})/(2\sqrt{n})$ where $H_{k}$ is the $k$-th Harmonic number and $\sigma_{\Omega}>0$, converges to the Continuum Random Tree $\mathcal{T}_{{\sf e}}$. Furthermore, we prove the local convergence of the rooted random $\Omega$-$k$-tree ${\sf G}_{n,k}^{\circ}$ to an infinite but locally finite random $\Omega$-$k$-tree ${\sf G}_{\infty,k}$.