We exhibit an interesting Cayley graph X of the elementary abelian group Z6 2 with the property that Aut(X) contains two regular subgroups, exactly one of which is normal. This demonstrates the existence of two subsets of Z6 2 that yield isomorphic Cayley graphs, even though the two subsets are not equivalent under the automorphism group of Z6 2 . 2000 Mathematics subject classification: 05C25.

We obtain a general bound on the Turán density of a hypergraph in terms of the number of edges that it contains. If F is an r-uniform hypergraph with f edges we show that π(F) < f−2 f−1 − (1 + o(1))(2r!2/rf3−2/r)−1, for fixed r ≥ 3 and f → ∞. Given an r-uniform hypergraph F , the Turán number of F is the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain a co...

Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and n is a k-uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for t ≥ 6. In this note we prove that for every k > t ≥ 2 and sufficiently large n,...

Let r be a finite connecied regular bipartite 2-arc transitive graph. It is shown that r is a cover of a possibly smaller graph :E, which is also connecied and regular of the same valency as r, and there is a subgroup G of Aut :E such that G is 2-arc transitive on :E and every nontrivial normal subgroup of G has at most two orbits on vertices. Such graphs :E for which the subgroup G has an abel...

A celebrated result in Ramsey Theory states that the order of magnitude of the trianglecomplete graph Ramsey numbers R(3, t) is t/ log t. In this paper, we consider an analogue of this problem for uniform hypergraphs. A triangle is a hypergraph consisting of edges e, f, g such that |e∩ f | = |f ∩ g| = |g ∩ e| = 1 and e ∩ f ∩ g = ∅. For all r ≥ 2, let R(C3,K t ) be the smallest positive integer ...

Claude BERGE C.N.R.S., Paris The hypergraphs whose chromatic number is ~ 2 ("bicolorable" hypergraphs) were introduced by E.W. Miller [13] under the name of "set-systems with Property B". This concept appears in Number Theory (see [5], [10]). It is also useful for some problems in positional games and Operations Research (see [3], [4], [7]); different results have been found under the form of i...

The modified bubble-sort graph of dimension n is the Cayley graph of Sn generated by n cyclically adjacent transpositions. In the present paper, it is shown that the automorphism group of the modified bubble sort graph of dimension n is Sn × D2n, for all n ≥ 5. Thus, a complete structural description of the automorphism group of the modified bubble-sort graph is obtained. A similar direct produ...

Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the maximum degree of a vertex of H. We conjecture that for every > 0, if D is sufficiently large as a function of t, k and , then the chromatic index of H is at most (t − 1 + 1/t + )D. We prove this conjecture for the special case of intersecting hypergraphs in the following stronger form:...

Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the maximum degree of a vertex of H. We conjecture that for every > 0, if D is sufficiently large as a function of t, k and , then the chromatic index of H is at most (t − 1 + 1/t + )D. We prove this conjecture for the special case of intersecting hypergraphs in the following stronger form:...

In this paper, we prove that all Cayley digraphs of valency 2 on nonabelian groups of odd square-free order are normal. For a given subset S of a finite group G without the identity element 1, the Cayley digraph on G with respect to S is denoted by r =Cay(G, S) where V(r) = G, E(r) = {(g, 8g) I 9 E G,8 E S}. It is clear that Aut (r), the automorphism group of r, contains the right regular repre...