159 research outputs found
S-Packing Colorings of Cubic Graphs
Given a non-decreasing sequence of positive
integers, an {\em -packing coloring} of a graph is a mapping from
to such that any two vertices with color
are at mutual distance greater than , . This paper
studies -packing colorings of (sub)cubic graphs. We prove that subcubic
graphs are -packing colorable and -packing
colorable. For subdivisions of subcubic graphs we derive sharper bounds, and we
provide an example of a cubic graph of order which is not
-packing colorable
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
A characterization of b-chromatic and partial Grundy numbers by induced subgraphs
Gy{\'a}rf{\'a}s et al. and Zaker have proven that the Grundy number of a
graph satisfies if and only if contains an induced
subgraph called a -atom.The family of -atoms has bounded order and
contains a finite number of graphs.In this article, we introduce equivalents of
-atoms for b-coloring and partial Grundy coloring.This concept is used to
prove that determining if and (under
conditions for the b-coloring), for a graph , is in XP with parameter .We
illustrate the utility of the concept of -atoms by giving results on
b-critical vertices and edges, on b-perfect graphs and on graphs of girth at
least
Subdivision into i-packings and S-packing chromatic number of some lattices
An -packing in a graph is a set of vertices at pairwise distance
greater than . For a nondecreasing sequence of integers
, the -packing chromatic number of a graph is
the least integer such that there exists a coloring of into colors
where each set of vertices colored , , is an -packing.
This paper describes various subdivisions of an -packing into -packings
(j\textgreater{}i) for the hexagonal, square and triangular lattices. These
results allow us to bound the -packing chromatic number for these graphs,
with more precise bounds and exact values for sequences ,
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
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
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
Approximation Algorithms for Multicoloring Planar Graphs and Powers of Square and Triangular Meshes
International audienceA multicoloring of a weighted graph G is an assignment of sets of colors to the vertices of G so that two adjacent vertices receive two disjoint sets of colors. A multicoloring problem on G is to find a multicoloring of G. In particular, we are interested in a minimum multicoloring that uses the least total number of colors. The main focus of this work is to obtain upper bounds on the weighted chromatic number of some classes of graphs in terms of the weighted clique number. We first propose an 11/6-approximation algorithm for multicoloring any weighted planar graph. We then study the multicoloring problem on powers of square and triangular meshes. Among other results, we show that the infinite triangular mesh is an induced subgraph of the fourth power of the infinite square mesh and we present 2-approximation algorithms for multicoloring a power square mesh and the second power of a triangular mesh, 3-approximation algorithms for multicoloring powers of semi-toroidal meshes and of triangular meshes and 4-approximation algorithm for multicoloring the power of a toroidal mesh. We also give similar algorithms for the Cartesian product of powers of paths and of cycles
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