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arXiv · 1308.3530

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

Takayuki Hibi·Aki Mori·Hidefumi Ohsugi·Akihiro Shikama
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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.

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Takayuki Hibi, Aki Mori, Hidefumi Ohsugi, Akihiro Shikama. 2013-08-16. The number of edges of the edge polytope of a finite simple graph. https://doi.org/10.26493/1855-3974.722.bba

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