Packings in bipartite prisms and hypercubes

Brešar, Boštjan (author), Klavžar, Sandi (author), Rall, Douglas F. (author)

The $2$-packing number $\rho_2(G)$ of a graph $G$ is the cardinality of a largest $2$-packing of $G$ and the open packing number $\rho^{\rm o}(G)$ is the cardinality of a largest open packing of $G$, where an open packing (resp. $2$-packing) is a set of vertices in $G$ no two (closed) neighborhoods of which intersect. It is proved that if $G$ is bipartite, then $\rho^{\rm o}(G\Box K_2) = 2\rho_2(G)$. For hypercubes, the lower bounds $\rho_2(Q_n) \ge 2^{n - \lfloor \log n\rfloor -1}$ and $\rho^{\rm o}(Q_n) \ge 2^{n - \lfloor \log (n-1)\rfloor -1}$ are established. These findings are applied to injective colorings of hypercubes. In particular, it is demonstrated that $Q_9$ is the smallest hypercube which is not perfect injectively colorable. It is also proved that $\gamma_t(Q_{2^k}\times H) = 2^{2^k-k}\gamma_t(H)$, where $H$ is an arbitrary graph with no isolated vertices.

• Language: English
• Keywords: 2-packing number, open packing number, bipartite prism, hypercube, injective coloring, total domination number
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: 6 str.
• Numbering: Vol. 347, iss. 4, art. 113875
• PID: 20.500.12556/RUL-154513
• UDC: 519.17
• ISSN on article: 0012-365X
• DOI: 10.1016/j.disc.2024.113875
• COBISS.SI-ID: 181387523
• Publication date in RUL: 19.02.2024

Record is a part of a journal

• Title: Discrete mathematics
• Shortened title: Discrete math.
• Publisher: Elsevier
• ISSN: 0012-365X
• COBISS.SI-ID: 1118479

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.

Secondary language

• Language: Slovenian
• Keywords: 2-pakirno število, odprto pakirno število, dvodelna prizma, hiperkocke, injektivno barvanje, celotno dominacijsko število