A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table

Hilton, A. J. W.; Spencer, C. L.

doi:10.1016/j.jcta.2009.02.001

Cited by 8 publications

(11 citation statements)

References 13 publications

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“…This result was extended in various ways [1,2,4,8,6]. One such extension that is directly relevant to us is given by Hilton and Spencer [10,11], showing that if G is the vertex-disjoint union of powers of cycles or of a power of a path and powers of cycles, then G is r-EKR (1 ≤ r ≤ α(G)), provided some condition on the clique number is satisfied (see [5] for short proofs with somewhat weaker bounds). The problem, however, of obtaining an EKR result for vertex-disjoint unions of (powers of) paths remained elusive.…”

Section: Introductionmentioning

confidence: 78%

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An Erdős-Ko-Rado Theorem for unions of length 2 paths

Feghali,

Hurlbert,

Kamat

2019

Preprint

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A family of sets is intersecting if any two sets in the family intersect. Given a graph G and an integer r ≥ 1, let I (r) (G) denote the family of independent sets of size r of G. For a vertex v of G, the family of independent sets of size r that contain v is called an r-star. Then G is said to be r-EKR if no intersecting subfamily of I (r) (G) is bigger than the largest r-star. Let n be a positive integer, and let G consist of the disjoint union of n paths each of length 2. We prove that if 1 ≤ r ≤ n/2, then G is r-EKR. This affirms a longstanding conjecture of Holroyd and Talbot for this class of graphs and can be seen as an analogue of a well-known theorem on signed sets, proved using different methods, by Deza and Frankl and by Bollobás and Leader.Our main approach is a novel probabilistic extension of Katona's elegant cycle method, which might be of independent interest.

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Section: Introductionmentioning

confidence: 78%

“…Example 2. Consider the parameter values in Figure 1, and let σ = (5,8,21,6,20,1,11,14,36,22,10,34,30,27,7,15,31,17,23,26,3,24,2,19,29,32,18,4,28,16,12,9,25,33,13,35).…”

Section: Preliminariesmentioning

confidence: 99%

An Erdős-Ko-Rado Theorem for unions of length 2 paths

Feghali,

Hurlbert,

Kamat

2019

Preprint

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“…Let A be an intersecting subfamily of I G (r) . Let g : V (G) → V (G) be the Talbot compression [20,29] given by…”

Section: P Borg and C Feghalimentioning

confidence: 99%

“…Inspired by the work of Talbot, Hilton and Spencer [19] went on to prove the following result, which is stated with notation used in [19,20].…”

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confidence: 99%

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The Hilton-Spencer cycle theorems via Katona's shadow intersection theorem

Borg¹,

Feghali²

2020

Discuss. Math. Graph Theory

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A family A of sets is said to be intersecting if every two sets in A intersect. An intersecting family is said to be trivial if its sets have a common element. A graph G is said to be r-EKR if at least one of the largest intersecting families of independent r-element sets of G is trivial. Let α(G) and ω(G) denote the independence number and the clique number of G, respectively. Hilton and Spencer recently showed that if G is the vertex-disjoint union of a cycle C raised to the power k and s cycles 1 C,. .. , s C raised to the powers k 1 ,. .. , k s , respectively, 1 ≤ r ≤ α(G), and min ω 1 C k1 ,. .. , ω s C ks ≥ ω C k , then G is r-EKR. They had shown that the same holds if C is replaced by a path P and the condition on the clique numbers is relaxed to min ω 1 C k1 ,. .. , ω s C ks ≥ ω P k. We use the classical Shadow Intersection Theorem of Katona to obtain a significantly shorter proof of each result for the case where the inequality for the minimum clique number is strict.

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Erdős–Ko–Rado theorems for a family of trees

Feghali

Johnson

Thomas

2018

Discrete Applied Mathematics

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Given a graph G and an integer r ≥ 1, let I (r) (G) denote the family of independent sets of size r of G. For a vertex v of G, let I (r) v (G) denote the family of independent sets of size r that contain v. This family is called an r-star and v is the centre of the star. Then G is said to be r-EKR if no pairwise intersecting subfamily of I (r) (G) is bigger than the largest r-star, and if every maximum size pairwise intersecting subfamily of I (r) (G) is an r-star, then G is said to be strictly r-EKR. Let µ(G) denote the minimum size of a maximal independent set of G. Holroyd and Talbot conjectured that if 2r ≤ µ(G), then G is r-EKR and strictly r-EKR if 2r < µ(G).An elongated claw is a tree in which one vertex is designated the root and no vertex other than the root has degree greater than 2. A depth-two claw is an elongated claw in which every vertex of degree 1 is at distance 2 from the root. We show that if G is a depth-two claw, then G is strictly r-EKR if 2r ≤ µ(G) + 1, confirming the conjecture of Holroyd and Talbot for this family. We also show that if G is an elongated claw with n leaves and at least one leaf adjacent to the root, then G is r-EKR if 2r ≤ n.Hurlbert and Kamat had conjectured that one can always find a largest r-star of a tree whose centre is a leaf. Baber and Borg have separately shown this to be false. We show that, moreover, for all n ≥ 2, d ≥ 3, there exists a positive integer r such that there is a tree where the centre of the largest r-star is a vertex of degree n at distance at least d from every leaf.

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A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table

Cited by 8 publications

References 13 publications

An Erdős-Ko-Rado Theorem for unions of length 2 paths

An Erdős-Ko-Rado Theorem for unions of length 2 paths

The Hilton-Spencer cycle theorems via Katona's shadow intersection theorem

Erdős–Ko–Rado theorems for a family of trees

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