Counting extensions revisited

TL;DR

This paper revisits the problem of counting specific rooted subgraphs in random graphs, providing new insights into the conditions under which these counts are asymptotically equal with high probability.

Contribution

It answers Spencer's open question by establishing the necessity of certain conditions for the asymptotic equality of extension counts in strictly balanced cases.

Findings

Necessary and sufficient conditions identified for extension count equality

Second moment method used to prove results

Discussion of open problems in the field

Abstract

We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. In 1989, Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices. For the important strictly balanced case, Spencer also raised the fundamental question as to whether these conditions are necessary. We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open.