Clustered colouring of graph products

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

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.

• Language: English
• Keywords: graph products, colouring, clustering
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2025
• Year: 2025
• Number of pages: 24 str.
• Numbering: Vol. 32, iss. 3, article no. P3.15
• PID: 20.500.12556/RUL-175142
• UDC: 519.17
• ISSN on article: 1077-8926
• DOI: 10.37236/13247
• COBISS.SI-ID: 253497859
• Publication date in RUL: 17.10.2025

Record is a part of a journal

• Title: The Electronic journal of combinatorics
• Shortened title: Electron. j. comb.
• Publisher: N.J. Calkin and H.S. Wilf
• ISSN: 1077-8926
• COBISS.SI-ID: 6973785

Licences

• License: CC BY-ND 4.0, Creative Commons Attribution-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nd/4.0/
• Description: Under the NoDerivatives Creative Commons license one can take a work released under this license and re-distribute it, but it cannot be shared with others in adapted form, and credit must be provided to the author.