On the number of labeled graphs of bounded treewidth

Abstract

Let Tn,k be the number of labeled graphs on n vertices and treewidth at most k (equivalently, the number of labeled partial k-trees). We show that ck2knlogkn2−k(k+3)2k−2k−2≤Tn,k≤k2knn2−k(k+1)2k−k,for k>1 and some explicit absolute constant c>0. Disregarding terms depending only on k, the gap between the lower and upper bound is of order (logk)n. The upper bound is a direct consequence of the well-known formula for the number of labeled k-trees, while the lower bound is obtained from an explicit construction. It follows from this construction that both bounds also apply to graphs of pathwidth and proper-pathwidth at most k.