Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Stéphane Perennes* Thème 1 — Réseaux et systèmes Projet SLOOP Rapport de recherche n ̊???? — Juillet 1996 — 25 pages Abstract: in this paper, we prove that the wrapped Butterfly digraph ~ WBF(d; n) of degree d and dimensionn contains at least d 1 arc-disjoint Hamilton circuits, answering a conjecture of D. Barth. We also conjecture that ~ WBF(d; ...

The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r − 1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r = 2, and very recently the conjecture was proved for the case where r = 3. It is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we p...

Let D be the triangle with an attached edge (i. e. D is the “kite”, a graph having vertices {a0, a1, a2, a3} and edges {a0, a1}, {a0, a2}, {a1, a2}, {a0, a3}). Bermond and Schönheim [6] proved that a kite-design of order n exists if and only if n ≡ 0 or 1 (mod 8). Let (W, C) be a nontrivial kite-design of order n ≥ 8, and let V ⊂ W with |V | = v < n. A path design (V,P) of order v and block siz...

The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r − 1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r = 2, and very recently the conjecture was proved for the case where r = 3. It is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we p...

As it is introduced by Bermond, Prennes, and Kodate and by Fragopoulou and Akl, some Cayley graphs, including most popular models for intercon-nection networks, admit a special automorphism, called complete rotation. Such an automorphism is often used to derive algorithms or properties of the underlying graph. For example, some optimal gossiping algorithms can be easily designed by using a comp...

A (simple undirected) graph G = (V,E) with m edges is graceful if it has a distinct vertex labeling f : V −→ {0, 1, 2, 3, . . . ,m} which induces a set of distinct edge labels {|f(u)− f(v)| | uv ∈ E, u, v ∈ V }. The famous Ringel-Kotzig conjecture [9, 15] is that all trees are graceful. The base of a tree T is obtained from T by deleting its one-degree vertices. A caterpillar is a tree whose ba...

For a 2-connected graph $G$ on $n$ vertices and two $x,y\in V(G)$, we prove that there is an $(x,y)$-path of length at least $k$ if are $\frac{n-1}{2}$ in $V(G)\backslash \{x,y\}$ degree $k$. This strengthens well-known theorem due to Erd\H{o}s Gallai 1959. As the first application this result, show with contains cycle $2k$ it has $\frac{n}{2}+k$ confirms 1975 conjecture made by Woodall. anothe...

Given a graph G, the problem is to determine an acyclic orientation of G which minimizes the maximal number of changes of orientation along any shortest path in G. The corresponding value is called the rank of the graph G. The motivation for this graph theoretical problem comes from the design of deadlock-free packet routing protocols [G. Tel, Deadlock-free packet switching networks, in: Introd...