HAVET , Frédéric

Un lieu pour comprendre l'informatique et pas juste consommer les produits numériques : c'est un rêve devenu projet, un projet devenu réalité. Voici Terra Numerica. Frédéric Havet et Dorian Mazauric vont nous expliquer en détails la démarche et partager quelques réalisations pour que nous aussi, parents, enseignant·e·s ou tout autre...

We consider on-board networks in satellites interconnecting entering signals (inputs) to amplifiers (outputs). The connections are made via expensive switches with four links available. The paths connecting inputs to outputs should be link-disjoint. Among the input signals, some of them, called priorities, must be connected to the amplifiers...

Laborde, Payan and Xuong conjectured that every digraph has a stable set meeting every longest path. We prove that this conjecture holds for digraphs with stability at most 2.

An (s,1)-total labelling of a graph G is an assignment of integers to V(G)E(G) such that: (i) any two adjacent vertices of G receive distinct integers, (ii) any two adjacent edges of G receive distinct integers, and (iii) incident vertex and edge receive integers that differ by at least s in absolute value. The span of a (s,1)-total labelling...

The Push Tree problem contains elements from both the Steiner Tree and Shortest Path problem. It deals with the trade-offs between the push and pull mechanism used in information distribution and retrieval. In , a two step approach for the Push Tree Problem was proposed. In the first step, a «good» spanning tree (called routing tree) is...

We first show that the choose number of the square of a subcubic graph with maximum average degree less than 18/7 is at most 6. As a corollary, we get that the choose number of the square of a planar graph with girth at least 9 is at most 6. We then show that the choose number of the square of a subcubic planar graph with girth at least 13 is at most 5.

An $(n,p,n+f)$-network $G$ is a graph $(V,E)$ where the vertex set $V$ is partitioned into four subsets $\calP$, $\calI$, $\calO$ and $\calS$ called respectively the priorities, the ordinary inputs, the outputs and the switches, satisfying the following constraints: there are $p$ priorities, $n-p$ ordinary inputs and $n+f$ outputs; each...

A tournament is an orientation of the edges of a complete graph. An arc is pancyclic in a digraph D if it is contained in a cycle of length l, for every $3\leq l\leq |D|$. In [4], Moon showed that every strong tournament contains at least three pancyclics arcs and characterized the tournaments with exactly three pancyclic arcs. All these...

We show that the choice number of the square of a subcubic graph with maximum average degree less than 18/7 is at most 6. As a corollary, we get that the choice number of the square of a subcubic planar graph with girth at least 9 is at most 6. We then show that the choice number of the square of a subcubic planar graph with girth at least 13...

A communication in a network is a pair of nodes (s, t). The node s is called the source source and t the destination. A communication set is a set of distinct communications, i.e. two communications might have the same source or the same destination, but they cannot have both same source and same destination. A routing of a communication (s, t)...

A communication in a network is a pair of nodes (s, t). The node s is called the source source and t the destination. A communication set is a set of distinct communications, i.e. two communications might have the same source or the same destination, but they cannot have both same source and same destination. A routing of a communication (s, t)...

Motivated by routing in telecommunication network using Software Defined Network (SDN) technologies, we consider the following problem of finding short routing lists using aggregation rules. We are given a set of communications X , which are distinct pairs (s, t) ⊆ S × T , (typically S is the set of sources and T the set of destinations), and a...

A good edge-labelling of a graph G is a labelling of its edges such that, for any ordered pair of vertices (x, y), there do not exist two paths from x to y with increasing labels. This notion was introduced in [2] to solve wavelength assignment problems for specific categories of graphs. In this paper, we aim at characterizing the class of...

A good edge-labelling of a graph G is a labelling of its edges such that for any two distinct vertices u, v, there is at most one (u, v)-path with non-decreasing labels. This notion was introduced in [3] to solve wavelength assignment problems for specific categories of graphs. In this paper, we aim at characterizing the class of graphs that...

A {\em good edge-labelling} of a graph $G$ is a labelling of its edges such that, for any ordered pair of vertices $(x,y)$, there do not exist two paths from $x$ to $y$ with increasing labels. This notion was introduced in~\cite{BCP} to solve wavelength assignment problems for specific categories of graphs. In this paper, we aim at...

A weighted orientation of a graph G is a pair (D, w) where D is an orientation of G and w is an arc-weighting of D, that is an application A(D) → N \ {0}. The in-weight of a vertex v in a weighted orientation (D, w), denoted by S (D,w) (v), is the sum of the weights of arcs with head v in D. A semi-proper orientation is a weighted orientation...

A proper vertex colouring of a graph $G$ is {\it 2-frugal} (resp. {\it linear}) if the graph induced by the vertices of any two colour classes is of maximum degree 2 (resp. is a forest of paths). A graph $G$ is {\it 2-frugally} (resp. {\it linearly}) {\it $L$-colourable} if for a given list assignment $L:V(G)\mapsto 2^{\mathbb N}$, there exists...

A {\it $(k_1,k_2)$-outdegree-splitting} of a digraph $D$ is a partition $(V_1,V_2)$ of its vertex set such that $D[V_1]$ and $D[V_2]$ have minimum outdegree at least $k_1$ and $k_2$, respectively. We show that there exists a minimum function $f_T$ such that every tournament of minimum outdegree at least $f_T(k_1,k_2)$ has a...

In this paper, we are interested in computing the number of edge colourings and total colourings of a connected graph. We prove that the maximum number of $k$-edge-colourings of a connected $k$-regular graph on $n$ vertices is $k\cdot((k-1)!)^{n/2}$. Our proof is constructive and leads to a branching algorithm enumerating all the...

A digraph is n-unavoidable if it is contained in every tournament of order n. We first prove that every arborescence of order n with k leaves is (n + k − 1)-unavoidable. We then prove that every oriented tree of order n with k leaves is (3 2 n + 3 2 k − 2)-unavoidable and (9 2 n − 5 2 k − 9 2)-unavoidable, and thus (21 8 (n − 1))-unavoidable....

Given a graph G = (V;E), two players, Alice and Bob, alternate their turns in choosing uncoloured vertices to be coloured. Whenever an uncoloured vertex is chosen, it is coloured by the least positive integer not used by any of its coloured neighbours. Alice's goal is to minimize the total number of colours used in the game, and Bob's goal is...

A {\it $(k_1,k_2)$-outdegree-splitting} of a digraph $D$ is a partition $(V_1,V_2)$ of its vertex set such that $D[V_1]$ and $D[V_2]$ have minimum outdegree at least $k_1$ and $k_2$, respectively. We show that there exists a minimum function $f_T$ such that every tournament of minimum outdegree at least $f_T(k_1,k_2)$ has a...

Ear decompositions of graphs are a standard concept related to several major problems in graph theory like the Traveling Salesman Problem. For example, the Hamiltonian Cycle Problem, which is notoriously N P-complete, is equivalent to deciding whether a given graph admits an ear decomposition in which all ears except one are trivial (i.e. of...

A {\it $(p,1)$-total labelling} of a graph $G=(V,E)$ is a total coloring $L$ from $V\cup E$ into $\{0,\dots ,l\}$ such that $|L(v)-L(e)|\geq p$ whenever an edge $e$ is incident to a vertex $v$. The minimum $l$ for which $G$ admits a $(p,1)$-total labelling is denoted by $\lambda_p(G)$. The case $p=1$ corresponds to the usual notion of total...

A natural generalization of graph colouring involves taking colours from a metric space and insisting that the endpoints of an edge receive colours separated by a minimum distance dictated by properties of the edge. In the $q$-backbone colouring problem, these minimum distances are either $q$ or $1$, depending on whether or not the edge is in...