Jean-Claude Bermond

Cycles in digraphs– a survey

The main subjects of this survey paper are Hamitonian cycles, cycles of prescirbed lengths, cycles in tournaments, and partitions, packings, and coverings by cycles. Several unsolved problems and a bibiligraphy are included.

Strategies for interconnection networks: some methods from graph theory

Abstract Interconnection networks require dense graphs in the sense that many nodes with relatively few links may be connected with relatively short paths. Some recent constructions of such dense graphs with a given maximal degree Δ and diameter D (known as (Δ, D ) graphs) are reviewed here. The paper also contains an updated table of the best known (Δ, D ) graphs.

Hamiltonian decomposition of Cayley graphs of degree 4

Abstract We prove that any 4-regular connected Cayley graph on a finite abelian group can be decomposed into two hamiltonian cycles. This answers a partial case of Alspachs conjecture concerning hamiltonian decompositions of 2k -regular connected Cayley graphs. As a corollary we obtain the hamiltonian decomposition of 2-jump circulant graphs, also called double loops.

Large fault-tolerant interconnection networks

This paper deals with reliability and fault-tolerant properties of networks. We first survey general reliability properties of networks, in particular those concerning diameter vulnerability. Then we study in details reliability properties of some families of networks in particular de Bruijn and Kautz networks and their generalizations which appear as very good fault-tolerant networks.

Large graphs with given degree and diameter. II

Abstract The following problem arises in the study of interconnection networks: find graphs of given maximum degree and diameter having the maximum number of vertices. Constructions based on a new product of graphs, which enable us to construct graphs of given maximum degree and diameter, having a great number of vertices from smaller ones are given; therefore the best values known before are improved considerably.

G-decomposition of Kn, where G has four vertices or less

Abstract The complete graph K n , is said to have a G -decomposition if it is the union of edge disjoint subgraphs each isomorphic to G . The set of values of n for which K n has a G -decomposition is determined if G has four vertices or less.

Efficient Collective Communication in Optical Networks

This paper studies the problems of broadcasting and gossiping in optical networks. In such networks the vast bandwidth available is utilized through wavelength division multiplexing: a single physical optical link can carry several logical signals, provided that they are transmitted on different wavelengths. In this paper we consider both single-hop and multihop optical networks. In single-hop networks the information, once transmitted as light, reaches its destination without being converted to electronic form in between, thus reaching high speed communication. In multihop networks a packet may have to be routed through a few intermediate nodes before reaching its final destination. In both models we give efficient broadcasting and gossiping algorithms, in terms of time and number of wavelengths. We consider both networks with arbitrary topologies and particular networks of practical interest. Several of our algorithms exhibit optimal performances.

Hardness and approximation of gathering in static radio networks

In this paper, we address the problem of gathering information in a central node of a radio network, where interference constraints are present. We take into account the fact that, when a node transmits, it produces interference in an area bigger than the area in which its message can actually be received. The network is modeled by a graph; a node is able to transmit one unit of information to the set of vertices at distance at most d/sub T/ in the graph, but when doing so it generates interference that does not allow nodes at distance up to d/sub I/ (d/sub I/ /spl ges/ d/sub T/) to listen to other transmissions. Time is synchronous and divided into time-steps in each of which a round (set of non-interfering radio transmissions) is performed. We give a general lower bound on the number of rounds required to gather on any graph, and present an algorithm working on any graph, with an approximation factor of 4. We also show that the problem of finding an optimal strategy for gathering (one that uses a minimum number of time-steps) does not admit a fully polynomial time approximation scheme if d/sub I/ > d/sub T/, unless P=NP, and in the case d/sub I/ = d/sub T/ the problem is NP-hard.

Traffic grooming in unidirectional WDM ring networks using design theory

We address the problem of traffic grooming in WDM rings with all-to-all uniform unitary traffic. We want to minimize the total number of SONET add-drop multiplexers (ADMs) required. We show that this problem corresponds to a partition of the edges of the complete graph into subgraphs, where each subgraph has at most C edges (where C is the grooming ratio) and where the total number of vertices has to be minimized. Using tools of graph and design theory, we optimally solve the problem for practical values and infinite congruence classes of values for a given C, and thus improve and unify all the preceding results. We disprove a conjecture of [A.L. Chiu and E.H. Modiano, 2000] saying that the minimum number of ADMs cannot be achieved with the minimum number of wavelengths and also another conjecture of [J.Q. Hu, 2002].

Sparse broadcast graphs

Abstract Broadcasting is an information dissemination process in which a message is to be sent from a single originator to all members of a network by placing calls over the communication lines of the network. Several previous papers have investigated ways to construct sparse graphs (networks) on n vertices in which this process can be completed in minimum time from any originator. In this paper, we describe four techniques to construct graphs of this type and show that they produce the sparsest known graphs for several values of n . For n = 18, n = 19, n = 30 and n = 31 we also show that our new graphs are minimum broadcast graphs (i.e., that no graph with fewer edges is possible). These new graphs can be used with other techniques to improve the best known results for many larger values of n .