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

Abstract

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