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### Extremal problems concerning Kneser graphs

P Frankl, Z Füredi
1986 Journal of combinatorial theory. Series B (Print)
For n 3 2k the vertex-set of the 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 &?  ...

### On the chromatic number of a random subgraph of the Kneser graph

S. G. Kiselev, A. M. Raigorodskii
We also discuss an interesting connection to an 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] .  ...

### On random subgraphs of Kneser and Schrijver graphs [article]

Andrey Borisovich Kupavskii
2015 arXiv   pre-print
A famous result due to Lovász states that the chromatic number of a 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  . Different questions concerning random graphs are discussed in the books  ,  .  ...

### 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.  ...

### A simple removal lemma for large nearly-intersecting families

Tuan Tran, Shagnik Das
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 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.  ...

### Independence and coloring properties of direct products of some vertex-transitive graphs

Mario Valencia-Pabon, Juan Vera
2006 Discrete Mathematics
Let (G) and (G) denote the independence number and chromatic number of a 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  (see also [6, 7] and  ...  Concerning homomorphisms between Kneser graphs, Stahl shows the following useful result. Theorem 5 (Stahl  ). Let m, n be integers such that n > 1 and m 2n.  ...

### Sharp bounds for the chromatic number of random Kneser graphs [article]

Sergei Kiselev, Andrey Kupavskii
2021 arXiv   pre-print
We also discuss an interesting connection to an 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.  ...

### Resolving sets for Johnson and Kneser graphs

Robert F. Bailey, José Cáceres, Delia Garijo, Antonio González, Alberto Márquez, Karen Meagher, María Luz Puertas
2013 European journal of combinatorics (Print)
In this paper, we consider the Johnson 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.  ...

### Removal and Stability for Erdős-Ko-Rado [article]

Shagnik Das, Tuan Tran
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 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.  ...

### Removal and Stability for Erdös--Ko--Rado

Shagnik Das, Tuan Tran
2016 SIAM Journal on Discrete Mathematics
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.  ...  One might be interested in the extremal problem with this stricter requirement, or in the case when k does not divide n.  ...

### 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  ...

### Extremal G-free induced subgraphs of Kneser graphs [article]

Meysam Alishahi, Ali Taherkhani
2018 arXiv   pre-print
The 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  ...

### 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 graph symmetry will swiftly move on to more advanced books such as Biggs  or Godsil and Royle  , 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.  ...

### 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  ...

### On the stability of the Erdős-Ko-Rado theorem [article]

Béla Bollobás, Bhargav Narayanan, Andrei Raigorodskii
2016 arXiv   pre-print
Delete the edges of a 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  who showed that an analogue of Mantel's Theorem is true for certain random graphs.  ...
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