Distance graphs with maximum chromatic number

Abstract : Let $D$ be a finite set of integers. The distance graph $G(D)$ has the set of integers as vertices and two vertices at distance $d ∈D$ are adjacent in $G(D)$. A conjecture of Xuding Zhu states that if the chromatic number of $G (D)$ achieves its maximum value $|D|+1$ then the graph has a clique of order $|D|$. We prove that the chromatic number of a distance graph with $D=\{ a,b,c,d\}$ is five if and only if either $D=\{1,2,3,4k\}$ or $D=\{ a,b,a+b,a+2b\}$ with $a \equiv 0 (mod 2)$ and $b \equiv 1 (mod 2)$. This confirms Zhu's conjecture for $|D|=4$.
Type de document :
Communication dans un congrès
Stefan Felsner. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), pp.171-174, 2005, DMTCS Proceedings
Liste complète des métadonnées

Littérature citée [11 références]  Voir  Masquer  Télécharger

https://hal.inria.fr/hal-01184347
Contributeur : Coordination Episciences Iam <>
Soumis le : vendredi 14 août 2015 - 11:36:36
Dernière modification le : jeudi 11 mai 2017 - 01:02:54
Document(s) archivé(s) le : dimanche 15 novembre 2015 - 10:57:28

Fichier

dmAE0134.pdf
Fichiers éditeurs autorisés sur une archive ouverte

Identifiants

  • HAL Id : hal-01184347, version 1

Collections

Citation

Javier Barajas, Oriol Serra. Distance graphs with maximum chromatic number. Stefan Felsner. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), pp.171-174, 2005, DMTCS Proceedings. 〈hal-01184347〉

Partager

Métriques

Consultations de la notice

57

Téléchargements de fichiers

222