The size of minimum 3-trees: cases 0 and 1 mod 12
نویسندگان
چکیده
A 3-uniform hypergraph is called a minimum 3-tree, if for any 3coloring of its vertex set there is a heterochromatic triple and the hypergraph has the minimum possible number of triples. There is a conjecture that the number of triples in such 3-tree is dn(n−2) 3 e for any number of vertices n. Here we give a proof of this conjecture for any n ≡ 0, 1mod 12.
منابع مشابه
The size of minimum 3-trees: Cases 3 and 4 mod 6
A 3-uniform hypergraph is called a minimum 3-tree if for any 3-coloring of its vertex set there is a heterochromatic edge and the hypergraph has the minimum possible numberof edges. Here we s h o w that the numberof edges in such 3-tree is l n(n;2) 3 m for any n umberof vertices n 3 4 (mod 6):
متن کاملFrames with Block Size Four
We investigate the spectrum for frames with block size four, and discuss several applications to the construction of other combinatorial designs. Our main result is that a frame of type hu, having blocks of size four, exists if and only if u ≥ 5, h ≡ 0 mod 3 and h(u − 1) ≡ 0 mod 4, except possibly where (i) h = 9 and u ∈ {13, 17, 29, 33, 93, 113, 133, 153, 173, 193}; (ii) h ≡ 0 mod 12 and u ∈ {...
متن کاملExistence of resolvable group divisible designs with block size four I
It is proved in this paper that for m ≡ 0; 2; 6; 10 (mod 12) there exists a resolvable group divisible design of order v, block size 4 and group size m if and only v ≡ 0 (mod 4), v ≡ 0 (modm), v − m ≡ 0 (mod 3), except when (3; 12) and except possibly when (3; 264); (3; 372); (8; 80); (8; 104), (9; 396) (40; 400) or (40; 520). c © 2002 Elsevier Science B.V. All rights reserved.
متن کاملOn the largest k th eigenvalues of trees with n ≡ 0 ( mod k ) ø
We consider the only remaining unsolved case n ≡ 0 (mod k) for the largest kth eigenvalue of trees with n vertices. In 1995, Jia-yu Shao gave complete solutions for the cases k = 2, 3, 4, 5, 6 and provided some necessary conditions for extremal trees in general cases (cf. [Linear Algebra Appl. 221 (1995) 131]). We further improve Shao’s necessary condition in this paper, which can be seen as th...
متن کاملBalanced incomplete block designs with block size~9: Part III
The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: λ(v − 1) ≡ 0 mod (k − 1) λv(v − 1) ≡ 0 mod k(k − 1) Earlier work has studied k = 9 with λ ∈ {1, 2, 3, 4, 6, 8, 9, 12}. In this article we show that the necessary conditions are sufficient for λ = 9 and every other λ not previously studied.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discussiones Mathematicae Graph Theory
دوره 23 شماره
صفحات -
تاریخ انتشار 2003