Dominique Sotteau

On the injective chromatic number of graphs

We define the concepts of an injective colouring and the injective chromatic number of a graph and give some upper and lower bounds in general, plus some exact values. We explore in particular the injective chromatic number of the hypercube and put it in the context of previous work on similar concepts, especially the theory of error-correcting codes. Finally, we give necessary, and sufficient conditions for the injective chromatic number te be equal to the degree for a regular graph.

Embeddings of complete binary trees into grids and extended grids with total vertex-congestion

Abstract Let G and H be two simple, undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H, together with a mapping which assigns to each edge [u,v] of G a path between f(u) and f(v) in H. The grid M(r,s) is the graph whose vertex set is the set of pairs on nonnegative integers, {(i,j) : 0⩽i 0⩽j , in which there is an edge between vertices (i,j) and (k,l) if either |i−k|=1 and j=l or i=k and |j−l|=1. The extended grid EM(r,s) is the graph whose vertex set is the set of pairs on nonnegative integers, {(i,j) : 0⩽i 0⩽j , in which there is an edge between vertices (i,j) and (k,l) if and only if |i−k|⩽1 and |j−l|⩽1. In this paper, we give embeddings of complete binary trees into square grids and extended grids with total vertex-congestion 1, i.e., for any vertex x of the extended grid we have load(x)+vertex-congestion(x)⩽1. Depending on the parity of the height of the tree, the expansion of these embeddings is approaching 1.606 or 1.51 for grids, and 1.208 or 1.247 for extended grids.

Cycles in the cube-connected cycles graph

In this paper we study the existence of cycles of all lengths in the cube-connected cycles graph and we establish that this graph is no far from being pancyclic in case n odd and bi-pancyclic in case n even.

Broadcasting and spanning trees in de Bruijin and Kautz networks

Abstract We prove that, for any p≤d, there exists a spanning directed p-ary tree of depth at most D⌈logpd⌉; in a de Bruijn digraph B(d,D) or in a Kautz digraph K(d,D) of degree d and diameter D. This result gives directly an upper bound of pD⌈logpd⌉ on the broadcast time of these digraphs, which improves the previously known bounds for d≥15. In the case of de Bruijn digraphs, an upper bound on the broadcast time of B(pq,D) in terms of the broadcast times of B(p,D) and B(q,D) is established. This is used to improve the upper bounds on the broadcast time of B(d,D). We obtain several results which are refinements of the following general statements: 1. (i) for any D⩾2, d⩾2, if 2 kl k , b(B(d, D))⩽( 5 4 k+3)D . 2. (ii) for any k⩾3, if 2k 1

Embedding Complete Binary Trees into Star and Pancake Graphs

Abstract. Let G and H be two simple undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H . The dilation of the embedding is the maximum distance between f(u),f(v) taken over all edges (u,v) of G . We give a construction of embeddings of dilation 1 of complete binary trees into star graphs. The height of the trees embedded with dilation 1 into the n -dimensional star graph is Ω (n log n) , which is asymptotically optimal. Constructions of embeddings of complete binary trees of dilation 2δ and 2δ +1 , δ≥ 1, into star graphs are given. The use of larger dilation allows embeddings of trees of greater height into star graphs. It is shown that all these constructions can be modified to yield embeddings of dilation 1 and 2δ , δ≥ 1 , of complete binary trees into pancake graphs.

Chemins Et Circuits Dans Les Graphes Orientes

Let D be a digraph with n vertices: we give sufficient conditions and conjectures on the number of arcs of D to insure that D has a directed cycle or a directed path of given length l , with more emphasis on the cases l = n, n - 1 or l small. We study the case where D is any digraph and the case where D is strong.

All-to-All Optical Routing in Chordal Rings of Degree 4

Abstract. We consider the problem of routing in networks employing all-optical routing technology. In such networks, information between nodes of the network is transmitted as light on fiber-optic lines without being converted to electronic form in between. We consider switched optical networks that use the wavelength-division multiplexing (or WDM) approach. A WDM network consists of nodes connected by point-to-point fiber-optic links, each of which can support a fixed number of wavelengths. The switches are capable of redirecting incoming streams based on wavelengths, without changing the wavelengths. Different messages may use the same link concurrently if they are assigned distinct wavelengths. However, messages assigned the same wavelength must be assigned edge-disjoint paths. Given a communication instance in a network, the optical routing problem is the assignment of {routes} to communication requests of the instance, as well as wavelengths to routes so that the number of wavelengths used by the instance is minimal. We focus on the all-to-all communication instance IA in a widely studied family of chordal rings of degree 4, called optimal chordal rings . For these networks, we prove exact bounds on the optimal load induced on an edge for IA , over all shortest-path routing schemes. We show an approximation algorithm that solves the optical routing problem for IA using at most 1.006 times the lower bound on the number of wavelengths. The previous best approximation algorithm has a performance ratio of 8. Furthermore, we use a variety of novel techniques to achieve this result, which are applicable to other communication instances and may be applicable to other networks.

2-diameter of de Bruijn graphs

This paper shows that in the undirected binary de Bruijn graph of dimension n, UB(n), which has diameter n, there exist at least two internally vertex disjoint paths of length at most n between any two vertices. In other words, the 2-diameter of UB(n) is equal to its diameter n.

Line graphs of hypergraphs I

Abstract We define the k -line graph of a hypergraph H as the graph whose vertices are the edges of H , two vertices being joined if the edges they represent intersect in at least k elements. In this paper we show that for any integer k and any graph G there exists a partial hypergraph H of some complete h -partite hypergraph K h h x N such that G is the k -line graph of H . We also prove that, for any integer p , there exist graphs which are not the ( h - p )-line graph of some h -uniform hypergraph. As a corollary we answer a problem of C. Cook. Further we show that it is not possible to characterize the ( h - 1)-line graphs by excluding a finite number of forbidden induced subgraphs.

Embedding complete binary trees into star networks

Star networks have been proposed as a possible interconnection network for massively parallel computers. In this paper we investigate embeddings of complete binary trees into star networks. Let G and H be two networks represented by simple undirected graphs. An embedding of G into H is an injective mapping f from the vertices of G into the vertices of H. The dilation of the embedding is the maximum distance between f(u), f(v) taken over all edges (u,v) of G.