The diamond-free process

TL;DR

This paper analyzes a random graph process that avoids creating a diamond graph (K_4^-), showing that the final graph size scales as (\u221a{\u2113}(n) (n^{3/2})), and that the resulting graph resembles a random graph with a triangle-free edge subset.

Contribution

It introduces and analyzes a new random graph process that avoids forming a diamond graph, providing asymptotic size and structural properties of the resulting graph.

Findings

Final graph size is ((({}(n))) (n^{3/2})) as n 

The process produces a graph resembling a random graph with a triangle-free edge subgraph

Edges not on triangles form a random-like subgraph

Abstract

Let K_4^- denote the diamond graph, formed by removing an edge from the complete graph K_4. We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of K_4^-. We show that, with probability tending to 1 as $n \to \infty$, the final size of the graph produced is $\Theta(\sqrt{\log(n)} \cdot n^{3/2})$. Our analysis also suggests that the graph produced after i edges are added resembles the random graph, with the additional condition that the edges which do not lie on triangles form a random-looking subgraph.