scholarly journals The Turán Polytope

10.37236/6555 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Annie Raymond
Keyword(s):  
Integer Program ◽  
Stable Set ◽  
Uniform Hypergraph ◽  
Stable Set Polytope ◽  
Turan's Theorem ◽  
Turán’S Theorem ◽  
Clique Inequalities ◽  
The Web

The Turán hypergraph problem asks to find the maximum number of $r$-edges in a $r$-uniform hypergraph on $n$ vertices that does not contain a clique of size $a$. When $r=2$, i.e., for graphs, the answer is well-known and can be found in Turán's theorem. However, when $r\ge 3$, the problem remains open. We model the problem as an integer program and call the underlying polytope the Turán polytope. We draw parallels between the latter and the stable set polytope: we show that generalized and transformed versions of the web and wheel inequalities are also facet-defining for the Turán polytope. We also show that clique inequalities and what we call doubling inequalities are facet-defining when $r=2$. These facets lead to a simple new polyhedral proof of Turán's theorem.

scholarly journals Clique-circulants and the stable set polytope of fuzzy circular interval graphs

2007 ◽  
Vol 115 (2) ◽  
pp. 291-317 ◽  
Author(s):  
Gianpaolo Oriolo ◽  
Gautier Stauffer
Keyword(s):  
Interval Graphs ◽  
Stable Set ◽  
Stable Set Polytope

scholarly journals On the Turán Properties of Infinite Graphs

10.37236/771 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Vojtěch Rödl
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
The Other ◽  
Other Hand ◽  
Turan's Theorem ◽  
Vertex Set ◽  
Turán’S Theorem ◽  
Positive Integers

Let $G^{(\infty)}$ be an infinite graph with the vertex set corresponding to the set of positive integers ${\Bbb N}$. Denote by $G^{(l)}$ a subgraph of $G^{(\infty)}$ which is spanned by the vertices $\{1,\dots,l\}$. As a possible extension of Turán's theorem to infinite graphs, in this paper we will examine how large $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$ can be for an infinite graph $G^{(\infty)}$, which does not contain an increasing path $I_k$ with $k+1$ vertices. We will show that for sufficiently large $k$ there are $I_k$–free infinite graphs with ${1\over 4}+{1\over 200} < \liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$. This disproves a conjecture of J. Czipszer, P. Erdős and A. Hajnal. On the other hand, we will show that $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}\le{1\over 3}$ for any $k$ and such $G^{(\infty)}$.

scholarly journals Turán’s Theorem for the Fano Plane

COMBINATORICA ◽  
2019 ◽  
Vol 39 (5) ◽  
pp. 961-982 ◽  
Author(s):  
Louis Bellmann ◽  
Christian Reiher
Keyword(s):  
Fano Plane ◽  
Turan's Theorem ◽  
Turán’S Theorem

A Strengthened General Cut-Generating Procedure for the Stable Set Polytope

2015 ◽  
Vol 50 ◽  
pp. 261-266 ◽  
Author(s):  
Ricardo C. Corrêa ◽  
Javier Marenco ◽  
Diego Delle Donne ◽  
Ivo Koch
Keyword(s):  
Stable Set ◽  
Stable Set Polytope

Turán's theorem in sparse random graphs

2003 ◽  
Vol 23 (3) ◽  
pp. 225-234 ◽  
Author(s):  
Tibor Szabó ◽  
Van H. Vu
Keyword(s):  
Random Graphs ◽  
Turan's Theorem ◽  
Turán’S Theorem

A Short Proof of Turán's Theorem

1999 ◽  
Vol 106 (3) ◽  
pp. 257-258
Author(s):  
William Staton
Keyword(s):  
Short Proof ◽  
Turan's Theorem ◽  
Turán’S Theorem
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