Partitionnement, recouvrement et colorabilité dans les graphes

Our research are about graph coloring with distance constraints (packing coloring) or neighborhood constraints (Grundy coloring). Let S={si| i in N*} be a non decreasing sequence of integers. An S-packing coloring is a proper coloring such that every set of color i is an si-packing (a set of vertices at pairwise distance greater than si). A graph G is (s1,... ,sk)-colorable if there exists a packing coloring of G with colors 1,... ,k. A Grundy coloring is a proper vertex coloring such that for every vertex of color i, u is adjacent to a vertex of color j, for each ji. These results allow us to determine S-packing coloring of these lattices for several sequences of integers. We examine a class of graph that has never been studied for S-packing coloring: the subcubic graphs. We determine that every subcubic graph is (1,2,2,2,2,2,2)-colorable and (1,1,2,2,3)-colorable. Few results are proven about some subclasses. Finally, we study the Grundy number of regular graphs. We determine a characterization of the cubic graphs with Grundy number 4. Moreover, we prove that every r-regular graph without induced square has Grundy number r+1, for ri. Ces résultats nous permettent de déterminer des S-colorations de packings de ces grilles pour plusieurs séries d’entiers. Nous examinons une classe de graphe jamais étudiée en ce qui concerne la S -coloration de packing: les graphes subcubiques. Nous déterminons que tous les graphes subcubiques sont (1,2,2,2,2,2,2)-colorables et (1,1,2,2,3)-colorables. Un certain nombre de résultats sont prouvés pour certaines sous-classes des graphes subcubiques. Pour finir, nous nous intéressons au nombre de Grundy des graphes réguliers. Nous déterminons une caractérisation des graphes cubiques avec un nombre de Grundy de 4. De plus, nous prouvons que tous les graphes r-réguliers sans carré induit ont pour nombre de Grundy de r+1, pour r<5

A Lower Bound for Radio kk-chromatic Number of an Arbitrary Graph

Radio $k$-coloring is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph $G$, subject to certain constraints involving the distance between the vertices. Specifically, for any simple connected graph $G$ with diameter $d$ and apositive integer $k$, $1\leq k \leq d$, a radio $k$-coloring of $G$ is an assignment $f$ of positive integers to the vertices of $G$ such that $|f(u)-f(v)|\geq 1+k-d(u, v)$, where $u$ and $v$ are any two distinct vertices of $G$ and $d(u, v)$ is the distance between $u$ and $v$.In this paper we give a lower bound for the radio $k$-chromatic number of an arbitrarygraph in terms of $k$, the total number of vertices $n$ and apositive integer $M$ such that $d(u,v)+d(v,w)+d(u,w)\leq M$ for all $u,v,w\in V(G)$. If $M$ is the triameter we get a better lower bound. We also find the triameter $M$ for several graphs, and show that the lower bound obtained for these graphs is sharp for the case $k=d$

Brooks' theorem on powers of graphs

We prove that for $k\geq 3$, the bound given by Brooks' theorem on the chromatic number of $k$-th powers of graphs of maximum degree $\Delta \geq 3$ can be lowered by 1, even in the case of online list coloring.Comment: 7 pages, no figure, submitte

Scalable partitioning for parallel position based dynamics

We introduce a practical partitioning technique designed for parallelizing Position Based Dynamics, and exploiting the ubiquitous multi-core processors present in current commodity GPUs. The input is a set of particles whose dynamics is influenced by spatial constraints. In the initialization phase, we build a graph in which each node corresponds to a constraint and two constraints are connected by an edge if they influence at least one common particle. We introduce a novel greedy algorithm for inserting additional constraints (phantoms) in the graph such that the resulting topology is q-colourable, where ˆ qˆ ≥ 2 is an arbitrary number. We color the graph, and the constraints with the same color are assigned to the same partition. Then, the set of constraints belonging to each partition is solved in parallel during the animation phase. We demonstrate this by using our partitioning technique; the performance hit caused by the GPU kernel calls is significantly decreased, leaving unaffected the visual quality, robustness and speed of serial position based dynamics

Random Constraint Satisfaction Problems

Random instances of constraint satisfaction problems such as k-SAT provide challenging benchmarks. If there are m constraints over n variables there is typically a large range of densities r=m/n where solutions are known to exist with probability close to one due to non-constructive arguments. However, no algorithms are known to find solutions efficiently with a non-vanishing probability at even much lower densities. This fact appears to be related to a phase transition in the set of all solutions. The goal of this extended abstract is to provide a perspective on this phenomenon, and on the computational challenge that it poses