A note on the least number of edges of 3-uniform hypergraphs with upper chromatic number 2
نویسندگان
چکیده
The upper chromatic number (H) of a hypergraphH= (X,E) is the maximum number k for which there exists a partition of X into non-empty subsets X=X1 ∪X2 ∪ · · · ∪Xk such that for each edge at least two vertices lie in one of the partite sets. We prove that for every n 3 there exists a 3-uniform hypergraph with n vertices, upper chromatic number 2 and n(n− 2)/3 edges which implies that a corresponding bound proved in [K. Diao, P. Zhao, H. Zhou, About the upper chromatic number of a co-hypergraph, Discrete Math. 220 (2000) 67–73] is best-possible. © 2006 Elsevier B.V. All rights reserved.
منابع مشابه
Uncolorable Mixed Hypergraphs
A mixed hypergraph H = (X; A; E) consists of the vertex set X and two families of subsets: the family E of edges and the family A of co-edges. In a coloring every edge E 2 E has at least two vertices of diierent colors, while every co-edge A 2 A has at least two vertices of the same color. The largest (smallest) number of colors for which there exists a coloring of a mixed hypergraph H using al...
متن کاملColloquia Mathematica Societatis Jan05 Bolyai 10. Infinite and Finite Sets, Keszthely (hungary), 1973 . Problems and Results on 3-chromatic Hypergraphs and Some Related Questions
A hypergraphi is a collection of sets. This paper deals with finite hy-pergraphs only. The sets in the hypergraph are called edges, the elements of these edges are points. The degree of a point is the number of edges containing it. The hypergraph is r-uniform if every edge has r points. A hypergraph is simple if any two edges have at most one common point, and it is called a clique if any two e...
متن کاملConstructions of sparse uniform hypergraphs with high chromatic number
Arandomconstruction gives newexamples of simple hypergraphswith high chromatic number that have few edges and/or lowmaximumdegree. In particular, for every integers k ≥ 2, r ≥ 2, and g ≥ 3, there exist r-uniform non-k-colorable hypergraphs of girth at least g with maximum degree at most r kr−1 ln k . This is only 4r2 ln k times greater than the lower bound by Erdős and Lovász (ColloquiaMath Soc...
متن کاملColoring H-free hypergraphs
Fix r ≥ 2 and a collection of r-uniform hypergraphs H. What is the minimum number of edges in an H-free r-uniform hypergraph with chromatic number greater than k? We investigate this question for various H. Our results include the following: • An (r, l)-system is an r-uniform hypergraph with every two edges sharing at most l vertices. For k sufficiently large, there is an (r, l)-system with chr...
متن کاملDense uniform hypergraphs have high list chromatic number
The first author showed that the list chromatic number of every graph with average degree d is at least (0.5 − o(1)) log2 d. We prove that for r ≥ 3, every r-uniform hypergraph in which at least half of the (r − 1)-vertex subsets are contained in at least d edges has list chromatic number at least ln d 100r3 . When r is fixed, this is sharp up to a constant factor.
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:
- Discrete Mathematics
دوره 306 شماره
صفحات -
تاریخ انتشار 2006