| Title: | Packings in bipartite prisms and hypercubes |
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| Authors: | ID Brešar, Boštjan (Author)
ID Klavžar, Sandi (Author)
ID Rall, Douglas F. (Author) |
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| Files: |  URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0012365X24000062
 PDF - Presentation file, download (231,57 KB)
MD5: BC4297894DBBF84BEE1923433A1D0D53
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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. |
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| Keywords: | 2-packing number, open packing number, bipartite prism, hypercube, injective coloring, total domination number |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.04.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | 6 str. |
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| Numbering: | Vol. 347, iss. 4, article no. 113875 |
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| PID: | 20.500.12556/DiRROS-18210  |
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| UDC: | 519.17 |
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| ISSN on article: | 0012-365X |
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| DOI: | 10.1016/j.disc.2024.113875 |
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| COBISS.SI-ID: | 181387523 |
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| Publication date in DiRROS: | 19.02.2024 |
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| Views: | 206 |
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| Downloads: | 76 |
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