On the b-continuity property of graphs
Identifieur interne : 004D46 ( Main/Exploration ); précédent : 004D45; suivant : 004D47On the b-continuity property of graphs
Auteurs : Dominique Barth [France] ; Johanne Cohen [France] ; Taoufik Faik [France]Source :
- Discrete applied mathematics [ 0166-218X ] ; 2007.
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- Pascal (Inist)
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Abstract
This paper deals with b-colorings of a graph G, that is, proper colorings in which for each color c, there exists at least one vertex colored by c such that its neighbors are colored by each other color. The b-chromatic number b(G) of a graph G is the maximum number of colors for which G has a b-coloring. It is easy to see that every graph G has a b-coloring using X(G) colors. We say that G is b-continuous iff for each k, X(G) ≤ k < b(G), there exists a b-coloring with k colors. It is well known that not all graphs are b-continuous. We call b-spectrum Sb(G) of G to be the set of integers k for which there is a b-coloring of G by k colors. We show that for any finite integer set I, there exists a graph whose b-spectrum is I and we investigate the complexity of the problem of deciding whether a graph G is b-continuous, even if b-colorings using X(G) and b(G) colors are given.
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aut.</sZ></inist:fA14><country>France</country><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Vandœuvre-lès-Nancy</settlement></placeName></affiliation></author><author><name sortKey="Faik, Taoufik" sort="Faik, Taoufik" uniqKey="Faik T" first="Taoufik" last="Faik">Taoufik Faik</name><affiliation wicri:level="3"><inist:fA14 i1="02"><s1>LORIA-CNRS, UMR 7503, Campus Scientifique, BP 239</s1><s2>54506 Vandoeuvre Les Nancy</s2><s3>FRA</s3><sZ>2 aut.</sZ><sZ>3 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Vandœuvre-lès-Nancy</settlement></placeName></affiliation></author></analytic><series><title level="j" type="main">Discrete applied mathematics</title><title level="j" type="abbreviated">Discrete appl. math.</title><idno type="ISSN">0166-218X</idno><imprint><date when="2007">2007</date></imprint></series></biblStruct></sourceDesc><seriesStmt><title level="j" type="main">Discrete applied mathematics</title><title level="j" type="abbreviated">Discrete appl. math.</title><idno type="ISSN">0166-218X</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Chromatic number</term><term>Color</term><term>Combinatorics</term><term>Complexity</term><term>Computer theory</term><term>Graph colouring</term><term>Integer</term><term>Maximum</term><term>Optimization</term><term>Spectrum</term><term>Vertex</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>Coloration graphe</term><term>Couleur</term><term>Vertex</term><term>Nombre chromatique</term><term>Maximum</term><term>Spectre</term><term>Nombre entier</term><term>Complexité</term><term>Informatique théorique</term><term>Optimisation</term><term>Combinatoire</term><term>68R10</term><term>Graphe coloré</term><term>05C15</term><term>47A10</term><term>Ensemble fini</term></keywords><keywords scheme="mix" xml:lang="en"><term>B-Chromatic number</term><term>Coloring</term><term>Complexity</term><term>Graph</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">This paper deals with b-colorings of a graph G, that is, proper colorings in which for each color c, there exists at least one vertex colored by c such that its neighbors are colored by each other color. The b-chromatic number b(G) of a graph G is the maximum number of colors for which G has a b-coloring. It is easy to see that every graph G has a b-coloring using X(G) colors. We say that G is b-continuous iff for each k, X(G) ≤ k < b(G), there exists a b-coloring with k colors. It is well known that not all graphs are b-continuous. We call b-spectrum S<sub>b</sub>(G) of G to be the set of integers k for which there is a b-coloring of G by k colors. We show that for any finite integer set I, there exists a graph whose b-spectrum is I and we investigate the complexity of the problem of deciding whether a graph G is b-continuous, even if b-colorings using X(G) and b(G) colors are given.</div></front></TEI><affiliations><list><country><li>France</li></country><region><li>Grand Est</li><li>Lorraine (région)</li><li>Île-de-France</li></region><settlement><li>Vandœuvre-lès-Nancy</li><li>Versailles</li></settlement></list><tree><country name="France"><region name="Île-de-France"><name sortKey="Barth, Dominique" sort="Barth, Dominique" uniqKey="Barth D" first="Dominique" last="Barth">Dominique Barth</name></region><name sortKey="Cohen, Johanne" sort="Cohen, Johanne" uniqKey="Cohen J" first="Johanne" last="Cohen">Johanne Cohen</name><name sortKey="Faik, Taoufik" sort="Faik, Taoufik" uniqKey="Faik T" first="Taoufik" last="Faik">Taoufik Faik</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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