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