Quelques problèmes de connexité dans les graphes orientés

AbstractWe prove that a minimally strongly h-connected digraph contains h + 1 vertices of half degree h. We study also the connectivity of transitive digraphs

Sur les parcours hamiltoniens dans les graphes orientes

AbstractWe prove that in a strongly h-connected digraph of order n ⩾ 8h3 + h there exists an Hamiltonian tour of length not exceeding Maxn+h2hn-hn-h2h,n+h2h∗n-hn-h2∗. This bound is best possible

Connectivite des graphes de cayley abeliens sans K4

AbstractWe give a description for the class of connected Abelian Cayley digraphs containing no K4 with connectivity less than the outdegree. We show that no member of this class is anti-symmetric

On the Subsets Product in Finite Groups

Let B be a proper subset of a finite group G such that either B = B−1 or G is abelian. We prove that there exists a subgroup H generated by an element of B with the following property. For every subset A of G such that A ∩ H ≠ ∅, either H ⊂ A ∪ AB or ❘A ∪ AB❘ , ❘A❘ + ❘B❘. This result generalizes the Cauchy-Davenport Theorem and two theorems of Chowla and Shepherdson

Topology of Cayley graphs applied to inverse additive problems

We present the basic isopermetric structure theory, obtaining some new simplified proofs. Let 1 ≤ r ≤ k be integers. As an application, we obtain simple descriptions for the subsets S of an abelian group with |kS| ≤ k|S|−k+1 or |kS−rS|−(k+r)|S|, where where S denotes as usual the Minkowski sum of copies of S. These results may be applied to several questions in Combinatorics and Additive Combinatorics including the Frobenius Problem, Waring’s problem in finite fields and the structure of abelian Cayley graphs with a big diameter.Peer Reviewe