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### Taylor Domination, Turán lemma, and Poincaré-Perron Sequences [article]

Dmitry Batenkov, Yosef Yomdin
2014 arXiv   pre-print
For f - rational function we show that Taylor domination is essentially equivalent to a well-known and widely used Turán's inequality on the sums of powers.  ...  This fact allows one to give a very short proof of Turán's lemma, albeit with a less sharp bound. Indeed, by Bezout theorem, rational functions of degree d are globally d-valent.  ...  ⊓ ⊔ Taylor domination in the maximal disk of convergence provided by Theorem 5.2, is not effective.  ...

### Dense neighbourhoods and Turán's theorem

Béla Bollobás, Andrew Thomason
1981 Journal of combinatorial theory. Series B (Print)
Straightforward reasoning shows that for every r/> 1 and n ~> 1 there is a unique r-partite graph of order n that has maximal size.  ...  We prove the following extension of Tur/m's theorem, conjectured by Erd6s.  ...  Consequently (4) holds if and only if G has the degree sequence of Tr(n) and equality holds in (3) .  ...

### On Turán's theorem for sparse graphs

M. Ajtai, P. Erdős, J. Komlós, E. Szemerédi
1981 Combinatorica
Proof of Theorem 2 The proof will use induction on n. We consider two cases according to the maximal valency .  ...  TURAN'S THEOREM FOR SPARSE GRAPHS 3 1 5 A sharper version of Theorem 1 A crucial point in the proof of Theorem 2 will be the application of the following sharper form of Theorem 1 : Theorem 1' .  ...

### Page 799 of Mathematical Reviews Vol. 58, Issue 2 [page]

1979 Mathematical Reviews
.; Nenov, Nedjalko D. 58 #5369 Extremal problems for s-graphs and Turan’s theorem. (Russian) Serdica 3 (1977), no. 2, 117-125. A classical theorem of P.  ...  holds for exactly those graphs for which it holds in Turan’s theorem).  ...

### Graphs with unavoidable subgraphs with large degrees

L. Caccetta, P. Erdös, K. Vijayan
1988 Journal of Graph Theory
For example, Turan's theorem gives a sufficient condition for G to contain a K"" in terms of the number of edges in G .  ...  In this paper we prove that, for m = an t , a > (k -1)/2k, G contains a K"" each vertex of which has degree at least f(a)n and determine the best possible f(a) .  ...  One of the best known results of this type is that of Turan 17 .81 : Turan's Theorem .  ...

### A hypergraph extension of Turán's theorem

Dhruv Mubayi
2006 Journal of combinatorial theory. Series B (Print)
Along the way, we give three proofs of a hypergraph generalization of Turán's theorem.  ...  We also prove a stability theorem for hypergraphs, analogous to the Simonovits stability theorem for complete graphs.  ...  . • Li and Li  proved Turán's theorem by looking at ideals in polynomials. This is perhaps the most striking and surprising proof of Turán's theorem.  ...

### An extension of Turán's Theorem, uniqueness and stability [article]

Peter Allen, Julia Böttcher, Jan Hladký, Diana Piguet
2014 arXiv   pre-print
Let us note that even though Theorem 2 extends Turán's theorem, the counting argument in Lemma 4 actually relies on Turán's result.  ...  In this case Turán's theorem guarantees that e(G) ≤ t r−1 (n) with equality if and only if G = T r−1 (n). Second, G contains at least one copy of K r .  ...

### Disjoint cliques in regular graphs of degree seven and eight

E.J Cockayne, S.T Hedetniemi
1978 Journal of combinatorial theory. Series B (Print)
A degree count shows 4 is not adjacent to ul, u2, and us. Hence 3 C(5, Z.Q , u2 , u,), contrary to maximality of y. Maximality of y, k implies each vertex of (H) has degree at least 3 in (H).  ...  Turan's theorem (see, e.g., [3, p. 2371) states that any graph with p vertices and T(p, n) edges contains K, where and p = t(n -1) + r (O<r<n-1) T(p, n) = (:) + 1 -t(p -n + 1 + r)/2. LEMMA 6.  ...

### Note

Stephan Brandt
2003 Combinatorica
In this note, a structural result for maximal Kr-free graphs is proven, which provides a simple proof of the Andrásfai-Erdős-Sós Theorem, saying that every Kr-free graph with minimum degree δ > (1 − 1  ...  In 1974, Andrásfai, Erdős and Sós  proved a lower bound on the minimum degree which ensures that a K r -free graph is (r − 1)-colourable: Theorem 1 (Andrásfai, Erdős and Sós ).  ...  From Proposition 2 we immediately get the proof of Theorem 1. Proof of Theorem 1. Let G be a K r -free graph. Choose a maximal K r -free supergraph G on V (G).  ...

### Quickly proving the Andrásfai-Erdős-Sós-Theorem [article]

Christian Reiher
2016 arXiv   pre-print
Given an integer r 2, an important theorem first proved by B. Andrásfai, P. Erdős, and V. T.  ...  Sós states that any K_r+1--free graph on n vertices whose minimum degree is greater than (3r-4)n/(3r-1) is r--colourable, and determines the graphs that are extremal in this context.  ...  As it turns out, that number is approximately r−1 2r · n 2 with an error of ±O r (1) and one way to show Turán's Theorem proceeds by repeatedly removing vertices of lowest degree.  ...

### Page 1222 of Mathematical Reviews Vol. , Issue 93c [page]

1993 Mathematical Reviews
This is exact for complete balanced w-partite graphs and gives Turan’s theorem when j = 1. A corollary is 8k} — 9k? + 3k34/ 16k} + 9k? ee 2 3 3 2 3 - 4k} — 18k?  ...  Math. 62 (1988), no. 3, 311-325; MR 89m:05086]: Theorem 1. Let H = (V,E) be a graph with maximum degree d, and let V = Vj; UK,U ---UV, bea partition of V into pairwise disjoint sets.  ...

### On Degrees in the Hasse Diagram of the Strong Bruhat Order [article]

2006 arXiv   pre-print
The maxima of the total degree and the down degree and their values at a random permutation are computed. Proofs involve variants of a classical theorem of Turán from extremal graph theory.  ...  For a permutation π in the symmetric group S_n let the total degree be its valency in the Hasse diagram of the strong Bruhat order on S_n, and let the down degree be the number of permutations which are  ...  Its name and certain other improvements were suggested by Christian Krattenthaler. Thanks also to Nathan Reading, Amitai Regev, Alexander Yong, and the anonymous referees.  ...

### Taylor Domination, Difference Equations, and Bautin Ideals [article]

Dmitry Batenkov, Yosef Yomdin
2014 arXiv   pre-print
We present some results and questions in this direction.  ...  This inequality is closely related to Turán's lemma (and to the "Turán third theorem" of [27, 28, 20] ).  ...  We consider a direct connection of Turán's lemma to Taylor domination, provided by Theorem 3.5 as an important and promising fact.  ...

### Cutwidth and degeneracy of graphs [article]

Benoit Kloeckner
2010 arXiv   pre-print
We prove an inequality involving the degeneracy, the cutwidth and the sparsity of graphs.  ...  It implies a quadratic lower bound on the cutwidth in terms of the degeneracy for all graphs and an improvement of it for clique-free graphs.  ...  Let G ′ be the δ(G)-core of G: its minimal degree is δ(G) and since it is a subgraph of G, cw(G) cw(G ′ ).  ...

### Page 60 of Mathematical Reviews Vol. 58, Issue 1 [page]

1979 Mathematical Reviews
It is shown that the only sequences of length 7 or more that can be realized as the degrees of a line-graph and not as a non-line-graph are those of the complete graph or those with maximal degree 2 or  ...  If for every integer k with 0<k<(n—q-—1)/2, the number of vertices of degree not exceed- ing k is less than k, and if k=(n—q —1)/2 for even (n—q-1), and the number of vertices of degree not exceeding k  ...
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