  A good edge-labeling of a graph [Ara\'ujo, Cohen, Giroire, Havet, Discrete
Appl. Math., forthcoming] is an assignment of numbers to the edges such that
for no pair of vertices, there exist two non-decreasing paths.
  In this paper, we study edge-labeling on graphs with girth at least 5. In
particular we verify, under this additional hypothesis, a conjecture by
Ara\'ujo et al. This conjecture states that if the average degree of G is less
than 3 and G is minimal without an edge-labeling, then G \in {C_3,K_{2,3}}.
(For the case when the girth is 4, we give a counterexample.)
