Maximum Treewidth of a Graphs with $m$ Edges What is the maximum treewidth of a graph with $m$ edges? In other 
words, what is the correct growth for the following function? 

$\alpha(m) = max\{\mathrm{treewidth}(G): G \mbox{ has $m$ edges}\}$. 

Clearly $\alpha(m) = \Omega(\sqrt{m})$, which is attained 
by cliques with $\Omega(\sqrt{m})$ vertices and by 
by $\Omega(\sqrt{m})\times \Omega(\sqrt{m})$ grids.

Is this also the maximum?
There are bounded degree (in fact even cubic i.e. 3-regular) expanders on $m$ vertices with treewidth at least $\epsilon \cdot m$, for some constant $\epsilon &gt; 0$. See Section 2 in:

Martin Grohe, Dániel Marx:
On tree width, bramble size, and expansion. J. Comb. Theory, Ser. B 99(1): 218-228 (2009)

Basically, they prove something much stronger: large vertex expansion is the only reason on why the treewidth can be as much as linear (in the number of vertices, but bounded degree graphs have linear number of edges).