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

Abstract

Let $d\ge 3$ 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 $3\le d\le 14$. 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.

arXiv comment10 pages, V2: Major Revision (We replace Section 2 with the results on the asymptotic behavior of ${\mu }_d$), V3-V4: The writing is improved

References in corpus (2)

• Simple polytopes arising from finite graphs
• Extremal edge polytopes

Cited by in corpus (2)

• Decomposable edge polytopes of finite graphs
• Faces of maximal chain polytopes