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

2014
*
arXiv
*
pre-print

For f - rational function we show that Taylor domination is essentially equivalent to a well-known

arXiv:1301.6033v2
fatcat:mepovytbgrguhkp77z3f2juw6y
*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. ...##
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Dense neighbourhoods and Turán's theorem

1981
*
Journal of combinatorial theory. Series B (Print)
*

Straightforward reasoning shows that for every r/> 1

doi:10.1016/s0095-8956(81)80016-0
fatcat:agwvbpfxbzglhhxny56dgejlcy
*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) . ...##
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On Turán's theorem for sparse graphs

1981
*
Combinatorica
*

Proof of

doi:10.1007/bf02579451
fatcat:vpi44ywa6vbztk6gjh7d7gkzue
*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' . ...##
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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*). ...##
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Graphs with unavoidable subgraphs with large degrees

1988
*
Journal of Graph Theory
*

For example,

doi:10.1002/jgt.3190120104
fatcat:lrxzqof5vffadoius7c4iwwgxa
*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

2006
*
Journal of combinatorial theory. Series B (Print)
*

Along the way, we give three proofs of a hypergraph generalization of

doi:10.1016/j.jctb.2005.06.013
fatcat:vefnju72yfaqla536igqsu6i5y
*Turán's**theorem*. ... We also prove a stability*theorem*for hypergraphs, analogous to the Simonovits stability*theorem*for complete graphs. ... . • Li*and*Li [16] proved*Turán's**theorem*by looking at ideals in polynomials. This is perhaps the most striking*and*surprising proof of*Turán's**theorem*. ...##
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An extension of Turán's Theorem, uniqueness and stability
[article]

2014
*
arXiv
*
pre-print

Let us note that even though

arXiv:1403.3801v2
fatcat:d6ff5iimmzevzpcz4vm7mvtjgi
*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

1978
*
Journal of combinatorial theory. Series B (Print)
*

A

doi:10.1016/0095-8956(78)90025-4
fatcat:77m73jnpcvfvxn6up7ibwfnyxu
*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. ...##
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Note

2003
*
Combinatorica
*

In this note, a structural result for

doi:10.1007/s00493-003-0042-z
fatcat:byez4hglkbgatlk64go7wi7v3q
*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 [1] 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). ...##
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Quickly proving the Andrásfai-Erdős-Sós-Theorem
[article]

2016
*
arXiv
*
pre-print

Given an integer r 2, an important

arXiv:1212.2521v2
fatcat:etzt4mb4erd4rhy6uxecvkoeqi
*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*. ...##
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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. ...##
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On Degrees in the Hasse Diagram of the Strong Bruhat Order
[article]

2006
*
arXiv
*
pre-print

The maxima of the total

arXiv:math/0505020v2
fatcat:gfmcokz6crfbzb5cbeojd4eqpy
*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. ...##
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Taylor Domination, Difference Equations, and Bautin Ideals
[article]

2014
*
arXiv
*
pre-print

We present some results

arXiv:1411.7629v1
fatcat:oyizzi7udbbc3o2uuqq6fgr3ue
*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. ...##
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Cutwidth and degeneracy of graphs
[article]

2010
*
arXiv
*
pre-print

We prove an inequality involving the degeneracy, the cutwidth

arXiv:0907.5138v2
fatcat:cmaab4akcrclnlyga27dkkweey
*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 ′ ). ...##
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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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