Treewidth is a graph parameter measuring how close a graph is to being a tree. I am interested in what is the minimal number of edges required for a graph to have treewidth $k$.
A natural family of graphs to consider are the cliques: a clique on $k+1$ vertices has treewidth $k$, and has $k (k+1) / 2$ edges. However, asymptotically, there are graphs of treewidth $k$ whose number of edges is linear rather than quadratic, e.g., cubic expanders as mentioned in this question. So cliques are not optimal. My question is: what is the smallest example of a graph of treewidth $k$ having less edges than the $(k+1)$-clique?
I have checked experimentally that all graphs of 14 edges or less have treewidth at most 4, so they do not beat the 6-clique which has 15 edges and treewidth 5. Hence, the smallest treewidth for which I do not know the answer is 6. Specifically: the 7-clique has 21 edges and treewidth 6, but is there a graph of treewidth 6 having 20 edges or less?
(Note that we could ask the same question of the minimal number of vertices required for a graph to have treewidth $k$, but then it is uninteresting: you need $k+1$ vertices for a graph to have treewidth $k$, and this is achieved by the $(k+1)$-clique.)
