scholarly journals An Extension of Turán's Theorem, Uniqueness and Stability

10.37236/4194 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Peter Allen ◽  
Julia Böttcher ◽  
Jan Hladký ◽  
Diana Piguet
Keyword(s):  
Extremal Graph ◽  
Extremal Graphs ◽  
Fixed Set ◽  
Vertex Graph ◽  
Turan's Theorem ◽  
Turán’S Theorem ◽  
Stability Results ◽  
Turán Graph

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$.  For $(r-1)|M|\ge n$, the $(r-1)$-partite Turán graph turns out to be the unique extremal graph. For $(r-1)|M|<n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.

scholarly journals Extremal Numbers for Odd Cycles

2014 ◽  
Vol 24 (4) ◽  
pp. 641-645 ◽  
Author(s):  
ZOLTAN FÜREDI ◽  
DAVID S. GUNDERSON
Keyword(s):  
Extremal Graph ◽  
Classical Theorem ◽  
Extremal Graphs ◽  
Complete Determination ◽  
Odd Cycles ◽  
Turán Graph

We describe theC2k+1-free graphs onnvertices with maximum number of edges. The extremal graphs are unique forn∉ {3k− 1, 3k, 4k− 2, 4k− 1}. The value ofex(n,C2k+1) can be read out from the works of Bondy [3], Woodall [14], and Bollobás [1], but here we give a new streamlined proof. The complete determination of the extremal graphs is also new.We obtain that the bound forn0(C2k+1) is 4kin the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turán graph.

scholarly journals Extremal Graph for Intersecting Odd Cycles

10.37236/5851 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Xinmin Hou ◽  
Yu Qiu ◽  
Boyuan Liu
Keyword(s):  
Complete Graph ◽  
Extremal Graph ◽  
Extremal Graphs ◽  
Turán Theorem ◽  
Odd Cycles ◽  
Turán Graph

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Turán graph, which is the complete $r$-partite graph on $n$ vertices with part sizes that differ by at most one. The well-known Turán Theorem states that $T_{n,r}$ is the only extremal graph for complete graph $K_{r+1}$. Erdős et al. (1995) determined the extremal graphs for intersecting triangles and Chen et al. (2003) determined the maximum number of edges of the extremal graphs for intersecting cliques. In this paper, we determine the extremal graphs for intersecting odd cycles.

scholarly journals On the Turán Number of Forests

10.37236/3142 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Hong Liu ◽  
Bernard Lidicky ◽  
Cory Palmer
Keyword(s):  
Arbitrary Order ◽  
Extremal Graph ◽  
Extremal Graphs ◽  
Turán Number

The Turán number of a graph $H$, $\mathrm{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. We determine the Turán number and find the unique extremal graph for forests consisting of paths when $n$ is sufficiently large. This generalizes a result of Bushaw and Kettle [Combinatorics, Probability and Computing 20:837--853, 2011]. We also determine the Turán number and extremal graphs for forests consisting of stars of arbitrary order.

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

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
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