Yahya Ould Hamidoune

On the Connectivity of Cayley Digraphs

We prove that the atom of a Cayley diagraph which contains the unity is a subgroup. As an application we obtain a short proof for a theorem of Imrich on the connectivity of the assignment polytope. We prove also that the connectivity of a Cayley digraph defined by a minimal generating set is equal to its indegree. This result generalizes a theorem of Godsil in the undirected case. We construct a class of Cayley digraphs with optimal connectivity. A symmetric subclass of our class was constructed by Boesch and Felger.

On the numbers of independent k -sets in a claw free graph

Abstract Let G be a graph without induced K1,3. The number of independent k-subsets of V(G) will be denoted by sk. We prove that the sequence (sk) is log concave and hence unimodal. These results are known for line graphs as a consequence of a result of Heilman and Lieb.

Representing a planar graph by vertical lines joining different levels

Answering a problem of H. de Fraysseix and P. Rosenstiehl we prove that every planar graph can be represented by horizontal segments corresponding to vertices and vertical segments corresponding to edges in such a way that no crossing appears. For 2-connected planar graphs, the boundary of the representation can be prescribed.

An Application of Connectivity Theory in Graphs to Factorizations of Elements in Groups

We obtain exact bounds for the girth and the diameter of a vertex-transitive digraph. As an application of the above results we obtain the following. Let G be a group and S be a generating subset such that 1 ∉ S , card(( G ) = n and card( S ) = s . Then the unity of G is the product of a sequence of elements of S with length not exceeding ⌈ n / s ⌉. If x is an arbitrary element of G then x is the product of a sequence of elements of G with length not exceeding (2 n −2 s −5)/ s . The bounds obtained in the paper are exact. We solve also the problem in the cases S = S −1 and S ∩ S −1 =∅.

Subsets with Small Sums in Abelian Groups' I

LetGbe an abelian group containing a finite subsetBsuch that, for every non-empty finite subsetA?G, |A+B|?min(|G|,|A|+|B|-1).We obtain the necessary and sufficient condition for the validity of the stronger property:For every finite subset A?G, such that |A|?2, |A+B|?min(|G|-1,|A|+|B|).We apply our methods to the range of diagonal forms over finite fields, obtaining a new proof of a result of Tietavainen. Our proof works in characteristic 2, where the question was open. We also apply our methods to obtain a new characterization for abelian Cayley graphs for which each minimum cutset originates or ends in a vertex.

On weighted sums in Abelian groups

Abstract Let G be an abelian group of order n and Davenport constant d and let k be a natural number. Let x 0 , x 1 , …, x m be a sequence of elements of G such that x o has the most repeated value in the sequence. Let { w i ; 1 ⩽ i ⩽ k } be a family of integers prime relative to n . We obtain the following two generalizations of the Erdos-Ginzburg-Ziv Theorem. For m ⩾ n + k − 1, we prove that there is a permutation α of [1, m ] such that ∑ 1 ⩽ i ⩽ k w i x α(i) = ∑ 1 ⩽ i ⩽ k w i x 0 . For k ⩾ n − 1 and m ⩾ k + d − 1, we prove that there is a k -subset K ⊂ [1, m ] such that ∑ i ∈ K x i =kx 0 .

Quelques problèmes de connexité dans les graphes orientés

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

Vosperian and superconnected Abelian Cayley digraphs

A digraphX is said to be Vosperian if any fragment has cardinality either 1 or|V(X)| − d+(X) − 1.A digraph is said to be superconnected if every minimum cutset is the set of vertices adjacent from or to some vertex.In this paper we characterize Vosperian and superconnected Abelian Cayley directed graphs. Our main tool is a difficult theorem of J.H. Kemperman from Additive Group Theory.In particular we characterize Vosperian and superconnected loops network (also called circulants).

Subsequence Sums

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On a Combinatorial Theorem of Erdös, Ginzburg and Ziv

G