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arXiv · 1211.2641

On the Maximum Density of Graphs with Good Edge-Labellings

Abbas Mehrabian·Dieter Mitsche·Paweł Prałat
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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.

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Abbas Mehrabian, Dieter Mitsche, Paweł Prałat. 2012-11-12. On the Maximum Density of Graphs with Good Edge-Labellings. https://arxiv.org/abs/1211.2641

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