On the density of nearly regular graphs with a good edge-labelling
  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. 
Abbas Mehrabian
math.CO