On the Maximum Density of Graphs with Good Edge-Labellings

TL;DR

This paper establishes a tight upper bound on the number of edges in graphs with good edge-labellings, showing such graphs have at most n log_2(n)/2 edges, and provides constructions matching this bound.

Contribution

It proves a new tight upper bound on edges in graphs with good edge-labellings and introduces a combinatorial lemma of independent interest.

Findings

Graphs with good edge-labellings have at most n log_2(n)/2 edges.

The bound is tight for infinitely many n.

Constructed graphs match the upper bound asymptotically.

Abstract

A good edge-labelling of a simple, finite graph is a labelling 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 good edge-labelling 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 good edge-labelling and has n log_2(n)/2 - O(n) edges.