Denise Amar

Irregularity strength of trees

Abstract The main result of this paper establishes that the irregularity strength of any tree with no vertices of degree two is its number of pendant vertices.

Cycles and paths of many lengths in bipartite digraphs

Abstract We give several sufficient conditions on the half-degrees of a bipartite digraph for the existence of cycles and paths of various lengths. Some analogous results are obtained for bipartite oriented graphs and for bipartite tournaments.

Pancyclism in hamiltonian graphs

We prove the following theorem. If G is a hamiltonian, nonbipartite graph of minimum degree at least (2n+1)5, where n represents the order of G, then G is pancyclic.

All-to-all wavelength-routing in all-optical compound networks

Abstract We give theoretical results obtained for wavelength routings in certain all-optical networks. In all-optical networks a single physical optical link can carry several logical signals provided that they are transmitted on different wavelengths. An all-to-all routing in a n -node network is a set of n(n−1) simple paths specified for every ordered pair (x,y) of nodes. The routing will be feasible if an assignment of wavelengths to the paths can be given such that no link will carry in the same direction two different paths of the routing on the same wavelength. With such a routing, it is possible to perform the gossiping in one round. The cost of the routing depends on the number of wavelengths it handles. We give the smallest necessary number of wavelength over all possible routings for networks based on certain compound graphs.

Pancyclism in Chvátal-Erdös's graphs

LetG = (X, E) be a simple graph of ordern, of stability numberα and of connectivityk withα ≤ k. The Chvátal-Erdöss theorem [3] proves thatG is hamiltonian. We have investigated under these conditions what can be said about the existence of cycles of lengthl. We have obtained several results:1)IfG ≠ Kk,k andG ≠ C5,G has aCn−1.2)IfG ≠ C5, the girth ofG is at most four.3)Ifα = 2 and ifG ≠ C4 orC5,G is pancyclic.4)Ifα = 3 and ifG ≠ K3,3,G has cycles of any length between four andn.5)IfG has noC3,G has aCn−2.

Cyclability and pancyclability in bipartite graphs

Abstract Let G be a 2-connected bipartite balanced graph of order 2 n and bipartition ( X , Y ). Let S be a subset of X of cardinality at least 3. We define S to be cyclable in G if there exists a cycle through all the vertices of S. Also, G is said S- pancyclable if for every integer l, 3⩽ l ⩽| S |, there exists a cycle in G that contains exactly l vertices of S. We prove that if the degree sum in G of every pair of nonadjacent vertices ( x , y ), x ∈ S , y ∈ Y is at least n +1, then S is cyclable in G. Under the same assumption where n +1 is replaced by n +3, we also prove that the graph G is S-pancyclable.

Biclosure and bistability in a balanced bipartite graph

The k-biclosure of a balanced bipartite graph wiht color classes A and B is the graph obtained from G by recursively joining pairs of nonadjacent vertices respectively taken in A and B whose degree sum is at least k, until no such pair remains. A property P defined on all the balanced bipartite graphs of order 2n is k-bistable if whenever G + ab has property P and dG(b) ≧ k then G itself has property P. We present a synthesis of results involving, for some properties, P, the bistability of P, the k-biclosure of G, the number of edges and the minimum degree.

Bipartite graphs with every matching in a cycle

We give sufficient Ore-type conditions for a balanced bipartite graph to contain every matching in a hamiltonian cycle or a cycle not necessarily hamiltonian. Moreover, for the hamiltonian case we prove that the condition is almost best possible.

Covering the vertices of a digraph by cycles of prescribed length

Abstract Let D be a strong digraph with n vertices and at least ( n − 1)( n − 2) + 3 arcs. For any integers k , n 1 , n 2 ,…, n k such that n = n 1 + n 2 +⋯+ n k and n i ⩾3, there exis covering of the vertices of D by disjoint directed cycles of length n 1 , n 2 ,…, n k except in two cases: 1. Case 1: n = 6; n 1 = n 2 = 3 and D contains a stable set with 3 vertices . 2. Case 2: n = 9; n 1 = n 2 = n 3 = 3 and D contains a stable set with 4 vertices .

Irregularity strength of regular graphs of large degree

Abstract As defined by Chartrand et al. [2], the irregularity strength of a graph G is the smallest possible value of k for which we can assign positive integers not greater than k to the edges of G in such a way that the sums at each vertex are distinct. We prove that, if G is an (n−3)- or (n−4)- regular graph of order n, the strength of G is 3 (except if G=K3,3); we conjecture that the irregularity strength of an r-regular graph of order n⩽2r is 3, except if G is Kl,l with l odd.