The number of edges of the edge polytope of a finite simple graph

TL;DR

This paper investigates the maximum number of edges in the edge polytope of finite simple graphs with a given number of vertices, establishing exact values for small sizes and exploring asymptotic behavior.

Contribution

It determines the exact maximum edge count for graphs with up to 14 vertices and improves existing lower bounds by constructing specific graph examples.

Findings

Maximum edges equals the complete graph case for 3 ≤ d ≤ 14

Established asymptotic behavior of maximum edge counts

Improved lower bounds using new graph constructions

Abstract

Let $d \geq 3$ be an integer. It is known that the number of edges of the edge polytope of the complete graph with $d$ vertices is $d(d-1)(d-2)/2$. In this paper, we study the maximum possible number $\mu_d$ of edges of the edge polytope arising from finite simple graphs with $d$ vertices. We show that $\mu_{d}=d(d-1)(d-2)/2$ if and only if $3 \leq d \leq 14$. In addition, we study the asymptotic behavior of $\mu_d$. Tran--Ziegler gave a lower bound for $\mu_d$ by constructing a random graph. We succeeded in improving this bound by constructing both a non-random graph and a random graph whose complement is bipartite.