A Cauchy-Davenport type result for arbitrary regular graphs

TL;DR

This paper extends a classical additive number theory result to regular graphs, establishing a dichotomy between small diameter and a large number of vertex pairs relative to edges.

Contribution

It proves a new theorem linking graph diameter and pair counts in regular graphs, inspired by the Cauchy-Davenport theorem for sumsets.

Findings

Either the graph has diameter at most 3 or the number of vertex pairs exceeds a constant multiple of edges.

Establishes a universal constant e > 0 for the dichotomy.

Provides a foundation for further research in graph structure and additive combinatorics.

Abstract

Motivated by the Cauchy-Davenport theorem for sumsets, and its interpretation in terms of Cayley graphs, we prove the following main result : There is a universal constant e > 0 such that, if G is a connected, regular graph on n vertices, then either every pair of vertices can be connected by a path of length at most 3, or the number of pairs of such vertices is at least 1+e times the number of edges in G. We discuss a range of further questions to which this result gives rise.