On the Maximum Density of Graphs with Unique-Path Labelings

A unique-path labeling of a simple, finite graph is a labeling of its edges with real numbers such that for every ordered pair of vertices $(u,v)$, there is at most one nondecreasing path from $u$ to $v$. In this paper we prove that any graph on $n$ vertices that admits a unique-path labeling has at most $n \log_2(n)/2$ edges and that this bound is tight for infinitely many values of $n$. Thus we significantly improve on the previously best known bounds. The main tool of the proof is a combinatorial lemma which might be of independent interest. For every $n$ we also construct an $n$-vertex graph that admits a unique-path labeling and has $n\log_2(n)/2 - O(n)$ edges.