Optimal packings of K4's into a Kn

Brouwer, AE Andries

doi:10.1016/0097-3165(79)90105-5

Cited by 131 publications

(168 citation statements)

References 10 publications

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“…Fixing one such forest P , w.l.o.g. the one defined by the edges (1, 2), (2, 3), (3,4), (5,6), (6, 7) (notice that this is indeed a labeled copy of W 7 ), we will prove that the probability that all the edges (π(1), π(2)), (π(3), π(4)), (π(5), π (6)) are colored the same, and that the edges (π(2), π(3)) and (π(6), π (7)) are also colored the same is less than 1/ (50n 2 q 7 ). This will show that the probability that Q π contains the weed W 7 is less than 1/(50n 2 ), as required.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…We note that the case H = K 3 is trivial since every properly edge-colored triangle is also rainbow colored. However, already for K 4 , the analogue of Brouwer's result [3] Let F be a family of positive integers. We say that K n is F -decomposable if we can color the edges of K n such that each color class induces a complete graph whose order belongs to F .…”

Section: Introductionmentioning

confidence: 99%

“…Such a system exists if and only if n mod 6 ∈ {1, 3} [9]. The case k = 4 is also known to exist if and only if n mod 12 ∈ {1, 4} [3].…”

Section: Introductionmentioning

confidence: 99%

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Rainbow decompositions

Yuster

2007

Proc. Amer. Math. Soc.

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A rainbow coloring of a graph is a coloring of the edges with distinct colors. We prove the following extension of Wilson's Theorem. For every integer k there exists an n 0 = n 0 (k) so that for all n > n 0 , if n mod k(k − 1) ∈ {1, k} then every properly edge-colored K n contains n 2 / k 2 pairwise edge-disjoint rainbow copies of K k .Our proof uses, as a main ingredient, a double application of the probabilistic method.

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rainbow decompositions

Yuster

2007

Proc. Amer. Math. Soc.

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“…Despite much effort, the exact value of T (n, F(k), 1) is known for all n only when k = 3 [14,15] and k = 4 [3]. Even for k = 5, there are an infinite number of n for which T (n, F (5), 1) is not yet determined.…”

Section: Introductionmentioning

confidence: 99%

“…F(3) contains only a single 3-graph, Λ(3, 2). Let (X, A) be a 3-graph of order n and size m containing exactly one copy of Λ (3,2). Then there exist exactly two blocks A, B ∈ A, with |A ∩ B| = 2.…”

Section: Introductionmentioning

confidence: 99%

On Extremal k-Graphs Without Repeated Copies of 2-Intersecting Edges

Chee¹,

Ling²

2007

SIAM J. Discrete Math.

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Abstract. The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turán numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k ∈ {3, 4}. We give a complete solution when k = 3 and an almost complete solution (with eleven exceptions) when k = 4.

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Constructions of optimal packing designs

Yin¹,

Assaf

1998

J. Combin. Designs

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Let v and k be positive integers. A (v, k, 1)‐packing design is an ordered pair (V, B) where V is a v‐set and B is a collection of k‐subsets of V (called blocks) such that every 2‐subset of V occurs in at most one block of B. The packing problem is mainly to determine the packing number P(k, v), that is, the maximum number of blocks in such a packing design. It is well known that P(k, v) ≤ ⌊v⌊(v − 1)/(k − 1)⌋/k⌋ = J(k, v) where ⌊×⌋ denotes the greatest integer y such that y ≤ x. A (v, k, 1)‐packing design having J(k, v) blocks is said to be optimal. In this article, we develop some general constructions to obtain optimal packing designs. As an application, we show that P(5, v) = J(5, v) if v ≡ 7, 11 or 15 (mod 20), with the exception of v ∈ {11, 15} and the possible exception of v ∈ {27, 47, 51, 67, 87, 135, 187, 231, 251, 291}. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 245–260, 1998

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Optimal packings of K4's into a Kn

Cited by 131 publications

References 10 publications

Rainbow decompositions

Rainbow decompositions

On Extremal k-Graphs Without Repeated Copies of 2-Intersecting Edges

Constructions of optimal packing designs

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