Edge-partitioning graphs into regular and locally irregular components

Abstract : A graph is locally irregular if every two adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs. They conjectured that for almost every graph $G$, there exists a minimum integer $\chi^{\prime}_{\mathrm{irr}}(G)$ such that $G$ admits an edge-partition into $\chi^{\prime}_{\mathrm{irr}}(G)$ classes, each of which induces a locally irregular graph. In particular, they conjectured that $\chi^{\prime}_{\mathrm{irr}}(G) \leq 3$ for every $G$, unless $G$ belongs to a well-characterized family of non-decomposable graphs. This conjecture is far from being settled, as notably (1) no constant upper bound on$\chi^{\prime}_{\mathrm{irr}}(G)$ is known for $G$ bipartite, and (2) no satisfactory general upper bound on $\chi^{\prime}_{\mathrm{irr}}(G)$ is known. We herein investigate the consequences on this question of allowing a decomposition to include regular components as well. As a main result, we prove that every bipartite graph admits such a decomposition into at most $6$ subgraphs. This result implies that every graph $G$ admits a decomposition into at most $6(\lfloor \mathrm{log} \chi (G) \rfloor +1)$ subgraphs whose components are regular or locally irregular.
Type de document :
Article dans une revue
Discrete Mathematics and Theoretical Computer Science, DMTCS, 2016, Vol. 17 no. 3 (3), pp.43-58
Liste complète des métadonnées

Littérature citée [6 références]  Voir  Masquer  Télécharger

https://hal.inria.fr/hal-01058019
Contributeur : Coordination Episciences Iam <>
Soumis le : mardi 16 août 2016 - 16:50:25
Dernière modification le : jeudi 7 septembre 2017 - 01:03:46
Document(s) archivé(s) le : jeudi 17 novembre 2016 - 10:34:13

Fichier

2770-9864-1-PB.pdf
Accord explicite pour ce dépôt

Identifiants

  • HAL Id : hal-01058019, version 3

Collections

Citation

Julien Bensmail, Brett Stevens. Edge-partitioning graphs into regular and locally irregular components. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2016, Vol. 17 no. 3 (3), pp.43-58. 〈hal-01058019v3〉

Partager

Métriques

Consultations de la notice

109

Téléchargements de fichiers

242