In this paper we determine some necessary conditions for a uniform hypergraph to have a given upper chromatic number. Mathenmatics Subject Classification: 05C05

We consider the problem of two-coloring n-uniform hypergraphs. It is known that any such hypergraph with at most 1 10 √

In ZF the existence of self-complementary graphs on every infinite set is equivalent to the statement that every infinite set is ’even’. We prove a generalization to k-uniform t-complementary hypergraphs. 1. Self-complementary graphs A graph G is called self-complementary if it is isomorphic to its complement G, which has the same vertices as G and exactly those edges that are not in G. First w...

We give a tight bound on randomized online coloring of hypergraphs. The bound holds even if the algorithm knows the hypergraph in advance (but not the ordering in which it is presented). More specifically, we show that for any n and k, there is a 2colorable k-uniform hypergraph on n vertices for which any randomized online coloring uses Ω(n/k) colors in expectation. 1 Online Hypergraph Coloring...

A k-uniform linear path of length l, denoted by P (k) l , is a family of k-sets {F1, . . . , Fl} such that |Fi ∩ Fi+1| = 1 for each i and Fi ∩ Fj = ∅ whenever |i − j| > 1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H , denoted by exk(n,H), is the maximum number of edges in a k-uniform hypergraph F on n vertices that does not contain H as a ...

We show that a simple variant of a naive colouring procedure can be used to list colour the edges of a k-uniform linear hypergraph of maximum degree provided every vertex has a list of at least +c(log) 4 1? 1 k available colours (where c is a constant which depends on k). We can extend this to colour hypergraphs of maximum codegree o(() with + o(() colours. This improves earlier results of Kahn...

In this paper, chromatic polynomials of (non-uniform) hypercycles, unicyclic hypergraphs, hypercacti and sunflower hypergraphs are presented. The formulae generalize known results for r-uniform hypergraphs due to Allagan, Borowiecki/ Lazuka, Dohmen and Tomescu. Furthermore, it is shown that the class of (non-uniform) hypertrees with m edges, where mr edges have size r, r ≥ 2, is chromatically c...

For d ≥ 2, let Hd(n, p) denote a random d-uniform hypergraph with n vertices in which each of the ( n d ) possible edges is present with probability p = p(n) independently, and let Hd(n,m) denote a uniformly distributed d-uniform hypergraph with n vertices and m edges. Let either H = Hd(n,m) or H = Hd(n, p), where m/n and ( n−1 d−1 ) p need to be bounded away from (d− 1)−1 and 0 respectively. W...

A classic result of G. A. Dirac in graph theory asserts that every n-vertex graph (n ≥ 3) with minimum degree at least n/2 contains a spanning (so-called Hamilton) cycle. G. Y. Katona and H. A. Kierstead suggested a possible extension of this result for k-uniform hypergraphs. There a Hamilton cycle of an n-vertex hypergraph corresponds to an ordering of the vertices such that every k consecutiv...