On the density of nearly regular graphs with a good edge-labelling

TL;DR

This paper investigates the structure of graphs with special edge-labelings that limit nondecreasing paths, showing that such 'good' graphs are sparse and providing bounds related to degree and girth.

Contribution

It establishes upper bounds on the density of good graphs with degree constraints and answers an open question about the existence of bad graphs with large girth.

Findings

Good graphs with degree close to average have at most n^{1+o(1)} edges.

Existence of bad graphs with arbitrarily large girth is demonstrated.

High girth graphs with bounded degree can be good, depending on girth size.

Abstract

A good edge-labelling of a simple graph is a labelling of its edges with real numbers such that, for any ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. Say a graph is good if it admits a good edge-labelling, and is bad otherwise. Our main result is that any good n-vertex graph whose maximum degree is within a constant factor of its average degree (in particular, any good regular graph) has at most n^{1+o(1)} edges. As a corollary, we show that there are bad graphs with arbitrarily large girth, answering a question of Bode, Farzad and Theis. We also prove that for any Delta, there is a g such that any graph with maximum degree at most Delta and girth at least g is good.