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Extremal problems concerning Kneser graphs

1986
*
Journal of combinatorial theory. Series B (Print)
*

For n 3 2k the vertex-set of the

doi:10.1016/0095-8956(86)90084-5
fatcat:idykn7nrvbckfbnnjapxaafucy
*Kneser**graph*K(n, k) is (f) and two vertices A, BE (t) are connected by an edge if A n B = 0. ... Each K is intersecting so this partition of (c) shows that the chromatic number of the*Kneser**graph*satisfies X(K(n, k)) < n -2k + 2. ... THE PROOF OF THEOREM 8 Let X be an r-*graph*on ZI elements (i.e., 1 u X 1 = u). As usual, denote by ex(n, X)=max(\F): Fc(~), JX( = n, F does not contain &? ...##
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On the chromatic number of a random subgraph of the Kneser graph

2017
*
Doklady. Mathematics
*

We also discuss an interesting connection to an

doi:10.1134/s1064562417050209
fatcat:2uc7afkbsbcexi64npoh24vfie
*extremal**problem*on embeddability of complexes. ... Given positive integers n ≥ 2k, a*Kneser**graph*KG n,k is a*graph*whose vertex set is the collection of all k-element subsets of the set {1, . . . , n}, with edges connecting pairs of disjoint sets. ... For results on the independence sets of*Kneser**graphs*and hypergraphs, see [11, 13, 14, 15, 16, 18, 24, 34] . The notion of the random*Kneser**graph*KG n,k (p) was introduced in [5, 6] . ...##
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On random subgraphs of Kneser and Schrijver graphs
[article]

2015
*
arXiv
*
pre-print

A famous result due to Lovász states that the chromatic number of a

arXiv:1502.00699v2
fatcat:jfjwzkjjebdejoye75qvqedlqe
*Kneser**graph*KG_n,k is equal to n-2k+2. ... Moreover, we obtain the same bounds on the chromatic numbers for the so-called Schrijver*graphs*, which are known to be vertex-critical induced subgraphs of*Kneser**graphs*. ... For a bit broader perspective on*extremal*questions for random*graphs*we refer the reader to the survey [14] . Different questions*concerning*random*graphs*are discussed in the books [1] , [4] . ...##
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Page 770 of Mathematical Reviews Vol. , Issue 2001B
[page]

2001
*
Mathematical Reviews
*

These results generalize known results for Z*, where Z,, is the n-path.”
2001b:05116 05C35
Rho, Yoomi (KR-SNU-GA; Seoul)
Erratum: “An

*extremal**problem**concerning**graphs*not containing K, and K,,,_,” [Discrete ... ) On the Prague dimension of*Kneser**graphs*. ...##
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A simple removal lemma for large nearly-intersecting families

2015
*
Electronic Notes in Discrete Mathematics
*

We then use this removal lemma to settle a question of Bollobás, Narayanan and Raigorodskii regarding the independence number of random subgraphs of the

doi:10.1016/j.endm.2015.06.015
fatcat:ru3yv5pg25hgxbzssl6lcqalcm
*Kneser**graph*K(n, k). ... The study of intersecting families is central to*extremal*set theory, dating back to the seminal Erdős-Ko-Rado theorem of 1961 that bounds the largest such families. ... To define the*problem*, we first need to introduce the*Kneser**graph*and its connection to the Erdős-Ko-Rado theorem. ...##
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Independence and coloring properties of direct products of some vertex-transitive graphs

2006
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Discrete Mathematics
*

Let (G) and (G) denote the independence number and chromatic number of a

doi:10.1016/j.disc.2006.04.013
fatcat:rb3sdjjhlfct5ln7ym6t7qozpe
*graph*G, respectively. Let ... One of the outstanding*problems*in*graph*theory is a formula*concerning*the chromatic number of the direct product of any two*graphs*G and H, called the Hedetniemi conjecture [8] (see also [6, 7] and ...*Concerning*homomorphisms between*Kneser**graphs*, Stahl shows the following useful result. Theorem 5 (Stahl [15] ). Let m, n be integers such that n > 1 and m 2n. ...##
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Sharp bounds for the chromatic number of random Kneser graphs
[article]

2021
*
arXiv
*
pre-print

We also discuss an interesting connection to an

arXiv:1810.01161v3
fatcat:qitoikqxerhitpglptqrvixtwu
*extremal**problem*on embeddability of complexes. ... Given positive integers n≥ 2k, the*Kneser**graph*KG_n,k is a*graph*whose vertex set is the collection of all k-element subsets of the set {1,..., n}, with edges connecting pairs of disjoint sets. ... We also related this*problem*to certain*extremal*properties of complexes. ...##
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Resolving sets for Johnson and Kneser graphs

2013
*
European journal of combinatorics (Print)
*

In this paper, we consider the Johnson

doi:10.1016/j.ejc.2012.10.008
fatcat:hn6tg2iinzgf3n623lm7exfpqe
*graphs*J(n,k) and*Kneser**graphs*K(n,k), and obtain various constructions of resolving sets for these*graphs*. ... designs, as well as (for*Kneser**graphs*) partial geometries, Hadamard matrices, Steiner systems and toroidal grids. ... In this paper, we are*concerned*with constructing resolving sets for Johnson and*Kneser**graphs*. ...##
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Removal and Stability for Erdős-Ko-Rado
[article]

2016
*
arXiv
*
pre-print

We use this removal lemma to settle a question of Bollobás, Narayanan and Raigorodskii regarding the independence number of random subgraphs of the

arXiv:1412.7885v3
fatcat:vzugstetpjf3tfnrdlzlyt7mpu
*Kneser**graph*K(n,k). ... The study of intersecting families is central to*extremal*set theory, dating back to the seminal Erdős-Ko-Rado theorem of 1961 that bounds the size of the largest such families. ... To define the*problem*at hand, we first need to introduce the*Kneser**graph*and its connection to the Erdős-Ko-Rado theorem. ...##
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Removal and Stability for Erdös--Ko--Rado

2016
*
SIAM Journal on Discrete Mathematics
*

The study of intersecting families is central to

doi:10.1137/15m105149x
fatcat:a3zc4gaxejf6fa42f7tzulmdga
*extremal*set theory, dating back to the seminal Erdős-Ko-Rado theorem of 1961 that bounds the size of the largest such families. ... To define the*problem*at hand, we first need to introduce the*Kneser**graph*and its connection to the Erdős-Ko-Rado theorem. ... One might be interested in the*extremal**problem*with this stricter requirement, or in the case when k does not divide n. ...##
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Page 4749 of Mathematical Reviews Vol. , Issue 87i
[page]

1987
*
Mathematical Reviews
*

(H-AOS)

*Extremal**problems**concerning**Kneser**graphs*. J. Combin. Theory Ser. B 40 (1986), no. 3, 270-284. Let e ) denote the set of all k-element subsets of an n-element set X. ... In this paper the authors study two closely related*problems*in*extremal**graph*theory. (1) Suppose that G is a*graph*on n vertices with minimum degree 6, G does not contain a complete subgraph on k vertices ...##
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Extremal G-free induced subgraphs of Kneser graphs
[article]

2018
*
arXiv
*
pre-print

The

arXiv:1801.03972v2
fatcat:kvdcdwintrhcxlr6o2pgtxss3q
*Kneser**graph*KG_n,k is a*graph*whose vertex set is the family of all k-subsets of [n] and two vertices are adjacent if their corresponding subsets are disjoint. ... to a given*graph*G. ... Another interesting generalization of the Erdős-Ko-Rado theorem can be found in [22, Theorem 3] which*concerns*the maximum number of vertices for a multipartite subgraph of the complement of*Kneser**graphs*...##
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Extremal Combinatorics: with Applications in Computer Science by Stasys Jukna, Springer, 2001, xvii + 375 pp. 32.50; $49.95, ISBN 3540663134

2004
*
Combinatorics, probability & computing
*

Most of the results presented in detail are fairly elementary: readers interested in

doi:10.1017/s096354830321244x
fatcat:7gh6o3yplzbbpco26gowr57pjq
*graph*symmetry will swiftly move on to more advanced books such as Biggs [4] or Godsil and Royle [5] , and the wide-ranging ... Several families of*graphs*are considered explicitly, such as generalized Petersen*graphs*,*Kneser**graphs*and metacirculant*graphs*. The last third of the book*concerns*reconstruction*problems*. ... The book contains eleven short chapters, seven*concerning**graph*automorphisms and related topics and four on reconstruction*problems*. ...##
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Page 46 of Mathematical Reviews Vol. , Issue 80A
[page]

1980
*
Mathematical Reviews
*

ISBN 0-12-111750-2 The author interprets

*extremal**graph*theory as “structural results and any relations among the invariants of a*graph*, especially those*concerned*with best possible inequalities”. ... Other highlights include a proof of the equivalence of the four color theorem and Hadwiger’s conjecture for 5-chromatic*graphs*, Mader’s results on minimally k-connected*graphs*, Lovdsz’ proof of the*Kneser*...##
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On the stability of the Erdős-Ko-Rado theorem
[article]

2016
*
arXiv
*
pre-print

Delete the edges of a

arXiv:1408.1288v2
fatcat:nwipvlw6lfbqteqtw5f4lvin5q
*Kneser**graph*independently of each other with some probability: for what probabilities is the independence number of this random*graph*equal to the independence number of the*Kneser*... Since an independent set in the*Kneser**graph*is the same as a uniform intersecting family, this gives us a random analogue of the Erdős-Ko-Rado theorem. ... Perhaps the first result of this kind in*extremal**graph*theory was proved by Babai, Simonovits, and Spencer [1] who showed that an analogue of Mantel's Theorem is true for certain random*graphs*. ...
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