We call a Cayley digraph X = Cay(G, 8) normal for G if the right regular representation R( G) of G is normal in the full automorphism group Aut(X) of X. In this paper we determine the normality of Cayley digraphs of groups of order twice a prime.

Let 2 ≤ q ≤ min{p, t − 1} be fixed and n → ∞. Suppose that F is a p-uniform hypergraph on n vertices that contains no complete q-uniform hypergraph on t vertices as a trace. We determine the asymptotic maximum size of F in many cases. For example, when q = 2 and p ∈ {t, t+ 1}, the maximum is ( n t−1 ) t−1 + o(nt−1), and when p = t = 3, it is b (n−1) 2 4 c for all n ≥ 3. Our proofs use the Krusk...

Let Γ = Cay(G, S) and G ≤ X ≤ AutΓ. We say Γ is (X, 1)-regular Cayley graph if X acts regularly on its arcs. Γ is said to be corefree if G is core-free in some X ≤ Aut(Cay(G, S)). In this paper, we prove that if an (X, 1)-regular Cayley graph of valency 5 is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free ...

where J is the all ones matrix and I is the identity. These matrix conditions are equivalent to the combinatorial conditions that the graph is both inand out-regular, and that the number of directed 2-paths from a vertex x to a vertex y is t if x = y, λ if x → y, and μ otherwise. Recently, these graphs were studied by Klin et al. [2], including some new constructions and a list of feasible para...

We consider strongly regular graphs r = (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman-Singleton graph, and the triangular graphs T(q) with q = 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We...

We show that almost every Cayley graph Γ of an abelian group G of odd prime-power order has automorphism group as small as possible. Additionally, we show that almost every Cayley (di)graph Γ of an abelian group G of odd prime-power order that does not have automorphism group as small as possible is a normal Cayley (di)graph of G (that is, GL/Aut(Γ)).

A conjecture of Erdős from 1965 suggests the minimum number of edges in a kuniform hypergraph on n vertices which forces a matching of size t, where t ≤ n/k. Our main result verifies this conjecture asymptotically, for all t < 0.48n/k. This gives an approximate answer to a question of Huang, Loh and Sudakov, who proved the conjecture for t ≤ n/3k. As a consequence of our result, we extend bound...