  A good edge-labelling of a simple graph is a labelling of its edges with real
numbers such that, for any ordered pair of vertices (u,v), there is at most one
nondecreasing path from u to v. Say a graph is good if it admits a good
edge-labelling, and is bad otherwise. Our main result is that any good n-vertex
graph whose maximum degree is within a constant factor of its average degree
(in particular, any good regular graph) has at most n^{1+o(1)} edges. As a
corollary, we show that there are bad graphs with arbitrarily large girth,
answering a question of Bode, Farzad and Theis. We also prove that for any
Delta, there is a g such that any graph with maximum degree at most Delta and
girth at least g is good.
