In this note we recall some well-known results about covering and k-covering of graphs. Then, we introduce a new notion of spanning tree which we call the p-spanning tree. This notion allows us to reduce the upper bound for the minimum integer k for which a connected graph G has no nontrivial k-covering. This note deals with the notion of covering of graphs and more precisely with the notion of...

We combine the theory of Coxeter groups, the covering theory of graphs introduced by Malnic, Nedela and Skoviera and the theory of reflections of graphs in order to obtain the following characterization of a Coxeter group: Let π : Γ → (v,D, ι,−1) be a 1-covering of a monopole admitting semiedges only. The graph Γ is the Cayley graph of a Coxeter group if and only if π is regular and any deck tr...

For a prime power q ≡ 1 (mod v), the q × q cyclotomic matrix, whose entries are the discrete logarithmsmodulo v of the entries in the addition table of Fq , has been shown using character theoretic arguments to produce an ε-biased array, provided that q is large enough as a function of v and ε. A suitable choice of ε ensures that the array is a covering array of strength t when q > t2v4t . On t...

1.1. Sites. We begin with the underlying idea for a Grothendieck topology. Consider your favorite topological space X. Its topology, don’t use the Zariski topology, is determined by the partially ordered category UX consisting of all the open subsets of X ordered with respect to inclusion. In order to verify that a presheaf is a sheaf on X, we must first construct all possible coverings U = (Ui...