Sparse pancyclic subgraphs of random graphs

TL;DR

This paper demonstrates that random graphs $G(n,p)$ with probability just above the pancyclicity threshold almost surely contain sparse subgraphs that are pancyclic, matching the minimal edge count known for complete graphs.

Contribution

It establishes the existence of sparse pancyclic subgraphs in random graphs near the pancyclicity threshold, extending known results from complete graphs to probabilistic models.

Findings

Random graphs $G(n,p)$ contain pancyclic subgraphs with minimal edges.

The threshold for $p$ is close to the known pancyclicity threshold.

Sparse pancyclic subgraphs exist with high probability near the threshold.

Abstract

It is known that the complete graph $K_n$ contains a pancyclic subgraph with $n+(1+o(1))\cdot \log _2 n$ edges, and that there is no pancyclic graph on $n$ vertices with fewer than $n+\log _2 (n-1) -1$ edges. We show that, with high probability, $G(n,p)$ contains a pancyclic subgraph with $n+(1+o(1))\log_2 n$ edges for $p \ge p^*$, where $p^*=(1+o(1))\ln n/n$, right above the threshold for pancyclicity.