ON THE DENSITY OF NEARLY REGULAR GRAPHS WITH A GOOD EDGE-LABELING

A good edge-labeling of a simple graph is a labeling 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-labeling, 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 [Good edge-labelings and graphs with girth at least five, preprint, available online at arXiv:1109.1125]. 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.