538 research outputs found
Graph multicoloring reduction methods and application to McDiarmid-Reed's Conjecture
A -coloring of a graph associates to each vertex a set of
colors from a set of colors in such a way that the color-sets of adjacent
vertices are disjoints. We define general reduction tools for -coloring
of graphs for . In particular, we prove necessary and sufficient
conditions for the existence of a -coloring of a path with prescribed
color-sets on its end-vertices. Other more complex -colorability
reductions are presented. The utility of these tools is exemplified on finite
triangle-free induced subgraphs of the triangular lattice. Computations on
millions of such graphs generated randomly show that our tools allow to find
(in linear time) a -coloring for each of them. Although there remain few
graphs for which our tools are not sufficient for finding a -coloring,
we believe that pursuing our method can lead to a solution of the conjecture of
McDiarmid-Reed.Comment: 27 page
Extended core and choosability of a graph
A graph is -choosable if for any color list of size associated
with each vertices, one can choose a subset of colors such that adjacent
vertices are colored with disjoint color sets. This paper shows an equivalence
between the -choosability of a graph and the -choosability of one
of its subgraphs called the extended core. As an application, this result
allows to prove the -choosability and -colorability of
triangle-free induced subgraphs of the triangular lattice.Comment: 10 page
Choosability of a weighted path and free-choosability of a cycle
A graph with a list of colors and weight for each vertex
is -colorable if one can choose a subset of colors from
for each vertex , such that adjacent vertices receive disjoint color
sets. In this paper, we give necessary and sufficient conditions for a weighted
path to be -colorable for some list assignments . Furthermore, we
solve the problem of the free-choosability of a cycle.Comment: 9 page
Vectorial solutions to list multicoloring problems on graphs
For a graph with a given list assignment on the vertices, we give an
algebraical description of the set of all weights such that is
-colorable, called permissible weights. Moreover, for a graph with a
given list and a given permissible weight , we describe the set of all
-colorings of . By the way, we solve the {\sl channel assignment
problem}. Furthermore, we describe the set of solutions to the {\sl on call
problem}: when is not a permissible weight, we find all the nearest
permissible weights . Finally, we give a solution to the non-recoloring
problem keeping a given subcoloring.Comment: 10 page
GEOM Module manual: I User guide
The GEOM module is part of the AMAPmod software and consists of a 3D objects description language. Based on the MTG model, this language provides a simple and ïŹexible mechanism to describe a hierarchical 3D scene as a collection of objects arranged into a graph structure, called Scene Graph. In addition to this module, AMAPmod includes a Viewer, which allow the user to examine the scenes he has created and to export them into various 3D ïŹle formats. This way it is possible to perform additional operations on the scenes such as ray tracing, walk through, hemispherical snapshots and so on. Although, this language has been designed to be used by non specialist and do not require strong backgrounds in 3D computer graphics, it is recommended to consult books introducing basic concepts on 3D graphics to have a better understanding. This document contains the following chapters: * The chapter 1 explains how to represent 3D scenes using AMAPmod. * The chapter 2 forms a reference to the GEOM's ïŹle formats. * The chapter 3 forms a reference to the objects available within the GEOM module
Du gĂšne Ă la fleur
National audienceAu-delà de l'expérimentation in vitro, l'expérimentation sur ordinateur devrait permettre de mieux comprendre les conditions de croissance des plantes
Quantifying the degree of self-nestedness of trees. Application to the structural analysis of plants
17 pagesInternational audienceIn this paper we are interested in the problem of approximating trees by trees with a particular self-nested structure. Self-nested trees are such that all their subtrees of a given height are isomorphic. We show that these trees present remarkable compression properties, with high compression rates. In order to measure how far a tree is from being a self-nested tree, we then study how to quantify the degree of self-nestedness of any tree. For this, we deïŹne a measure of the self-nestedness of a tree by constructing a self-nested tree that minimizes the distance of the original tree to the set of self-nested trees that embed the initial tree. We show that this measure can be computed in polynomial time and depict the corresponding algorithm. The distance to this nearest embedding self-nested tree (NEST) is then used to deïŹne compression coefïŹcients that reïŹect the compressibility of a tree. To illustrate this approach, we then apply these notions to the analysis of plant branching structures. Based on a database of simulated theoretical plants in which different levels of noise have been introduced, we evaluate the method and show that the NESTs of such branching structures restore partly or completely the original, noiseless, branching structures. The whole approach is then applied to the analysis of a real plant (a rice panicle) whose topological structure was completely measured. We show that the NEST of this plant may be interpreted in biological terms and may be used to reveal important aspects of the plant growth
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