On the Maximum Density of Graphs with Good Edge-Labellings
Combinatorics
2014-03-18 v1
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.
Keywords
Cite
@article{arxiv.1211.2641,
title = {On the Maximum Density of Graphs with Good Edge-Labellings},
author = {Abbas Mehrabian and Dieter Mitsche and Paweł Prałat},
journal= {arXiv preprint arXiv:1211.2641},
year = {2014}
}
Comments
10 pages