Injective colorings of Sierpiński-like graphs and Kneser graphs

Brešar, Boštjan (author), Klavžar, Sandi (author), Samadi, Babak (author), Yero, Ismael G. (author)

Two relationships between the injective chromatic number and, respectively, chromatic number and chromatic index, are proved. They are applied to determine the injective chromatic number of Sierpiński graphs and to give a short proof that Sierpiński graphs are Class 1. Sierpiński-like graphs are also considered, including generalized Sierpiński graphs over cycles and rooted products. It is proved that the injective chromatic number of a rooted product of two graphs lies in a set of six possible values. Sierpiński graphs and Kneser graphs $K(n,r)$ are considered with respect of being perfect injectively colorable, where a graph is perfect injectively colorable if it has an injective coloring in which every color class forms an open packing of largest cardinality. In particular, all Sierpiński graphs and Kneser graphs $K(n,r)$ with $n\ge 3r-1$, are perfect injectively colorable, while $K(7,3)$ is not.

• Language: English
• Keywords: injective coloring, injective chromatic number, perfect injectively colorable graph, Sierpiński graphs, Kneser graphs, rooted product graph
• 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: 01.08.2025
• Year: 2025
• Number of pages: 22 str.
• Numbering: Vol. 41, iss. 4, art. no. 83
• PID: 20.500.12556/RUL-170988
• UDC: 519.17
• ISSN on article: 0911-0119
• DOI: 10.1007/s00373-025-02952-3
• COBISS.SI-ID: 243802627
• Publication date in RUL: 25.07.2025

Record is a part of a journal

• Title: Graphs and combinatorics
• Shortened title: Graphs comb.
• Publisher: Springer
• ISSN: 0911-0119
• COBISS.SI-ID: 25536512

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: injektivno barvanje, injektivno kromatično število, popolno injektivno obarljiv graf, grafi Sierpińskega, Kneserjevi grafi, korenski produkt grafov