Clustered colouring of graph products

Campbell, Rutger (avtor), Gollin, J. Pascal (avtor), Hendrey, Kevin (avtor), Lesgourgues, Thomas (avtor), Mohar, Bojan (avtor), Tamitegama, Youri (avtor), Tan, Jane (avtor), Wood, David Richard (avtor)

A colouring of a graph $G$ has clustering $k$ if the maximum number of vertices in a monochromatic component equals $k$. Motivated by recent results showing that many natural graph classes are subgraphs of the strong product of a graph with bounded treewidth and a path, this paper studies clustered colouring of strong products of two bounded treewidth graphs, where none, one, or both graphs have bounded degree. For example, in the case of two colours, if n is the number of vertices in the product, then we show that clustering $\Theta(n^{2/3})$ is best possible, even if one of the graphs is a path. However, if both graphs have bounded degree, then clustering $\Theta(n^{1/2})$ is best possible. With three colours, if one of the graphs has bounded degree, then we show that clustering $\Theta(n^{3/7})$ is best possible. However, if neither graph has bounded degree, then clustering $\Omega(n^{1/2})$ is necessary. More general bounds for any given number of colours are also presented.

• Jezik: Angleški jezik
• Ključne besede: graph products, colouring, clustering
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Datum objave: 01.01.2025
• Leto izida: 2025
• Št. strani: 24 str.
• Številčenje: Vol. 32, iss. 3, article no. P3.15
• PID: 20.500.12556/RUL-175142
• UDK: 519.17
• ISSN pri članku: 1077-8926
• DOI: 10.37236/13247
• COBISS.SI-ID: 253497859
• Datum objave v RUL: 17.10.2025

Gradivo je del revije

• Naslov: The Electronic journal of combinatorics
• Skrajšan naslov: Electron. j. comb.
• Založnik: N.J. Calkin and H.S. Wilf
• ISSN: 1077-8926
• COBISS.SI-ID: 6973785

Licence

• Licenca: CC BY-ND 4.0, Creative Commons Priznanje avtorstva-Brez predelav 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by-nd/4.0/deed.sl
• Opis: Licenca Creative Commons Brez predelav dovoljuje uporabnikom ponovno distribucijo dela, vendar ne v spremenjeni obliki. Zahtevana je navedba avtorstva.