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

For any set Ω of non-negative integers such that { 0 , 1 } ⊊ Ω , we consider a random Ω-k-tree G n,k that is uniformly selected from all connected k-trees of (n + k) vertices such that the number of (k + 1)-cliques that contain any fixed k-clique belongs to Ω. We prove that Gn,k, scaled by ( k H k σ Ω ) / ( 2 n ) where H k is the kth harmonic number and σ Ω > 0, converges to the continuum random tree T e . Furthermore, we prove local convergence of the random Ω-k-tree G n , k ∘ to an infinite but locally finite random Ω-k-tree G∞,k.