Covers in Uniform Intersecting Families and a Counterexample to a Conjecture of Lovász
نویسندگان
چکیده
We discuss the maximum size of uniform intersecting families with covering number at least . Among others, we construct a large k-uniform intersecting family with covering number k, which provides a counterexample to a conjecture of Lov asz. The construction for odd k can be visualized on an annulus, while for even k on a Mobius band.
منابع مشابه
On the oriented perfect path double cover conjecture
An oriented perfect path double cover (OPPDC) of a graph $G$ is a collection of directed paths in the symmetric orientation $G_s$ of $G$ such that each arc of $G_s$ lies in exactly one of the paths and each vertex of $G$ appears just once as a beginning and just once as an end of a path. Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete Math. 276 (2004) 287-294) conjectured that ...
متن کاملUniform Intersecting Families with Covering Number Restrictions
It is known that any k-uniform family with covering number t has at most k t t-covers. In this paper, we give better upper bounds for the number of t-covers in k-uniform intersecting families with covering number t.
متن کاملA New Construction of Non-Extendable Intersecting Families of Sets
In 1975, Lovász conjectured that any maximal intersecting family of k-sets has at most b(e− 1)k!c blocks, where e is the base of the natural logarithm. This conjecture was disproved in 1996 by Frankl and his co-authors. In this short note, we reprove the result of Frankl et al. using a vastly simplified construction of maximal intersecting families with many blocks. This construction yields a m...
متن کاملCovers in 5-uniform intersecting families with covering number three
Let k be an integer. It is known that the maximum number of threecovers of a k-uniform intersecting family with covering number three is k − 3k + 6k − 4 for k = 3, 4 or k ≥ 9. In this paper, we prove that the same holds for k = 5, and show that a 5-uniform intersecting family with covering number three which has 76 three-covers is uniquely determined.
متن کاملFractional aspects of the Erdös-Faber-Lovász Conjecture
The Erdős-Faber-Lovász conjecture is the statement that every graph that is the union of n cliques of size n intersecting pairwise in at most one vertex has chromatic number n. Kahn and Seymour proved a fractional version of this conjecture, where the chromatic number is replaced by the fractional chromatic number. In this note we investigate similar fractional relaxations of the Erdős-Faber-Lo...
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:
- J. Comb. Theory, Ser. A
دوره 74 شماره
صفحات -
تاریخ انتشار 1996