| Title: | The k-independence number of direct products of graphs and Hedetniemi's conjecture |
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| Authors: | ID Špacapan, Simon (Author) |
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| Files: |  http://dx.doi.org/10.1016/j.ejc.2011.07.002
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| Language: | English |
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| Work type: | Not categorized |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | FS - Faculty of Mechanical Engineering
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| Abstract: | The ▫$k$▫-independence number of ▫$G$▫, denoted as ▫$alpha_k(G)$▫, is the size of a largest ▫$k$▫-colorable subgraph of ▫$G$▫. The direct product of graphs ▫$G$▫ and ▫$H$▫, denoted as ▫$G times H$▫, is the graph with vertex set ▫$V(G) times V(H)$▫, where two vertices ▫$(x_1, y_1)$▫ and ▫$(x_2, y_2)$▫ are adjacent in ▫$G times H$▫, if ▫$x_1$▫ is adjacent to ▫$x_2$▫ in ▫$G$▫ and ▫$y_1$▫ is adjacent to ▫$y_2$▫ in ▫$H$▫. We conjecture that for any graphs ▫$G$▫ and ▫$H$▫, ▫$$alpha_k(G times H) ge alpha_k(G)|V(H)| + alpha_k(H)|V(G)| - alpha_k(G) alpha_k(H).$$▫ The conjecture is stronger than Hedetniemi's conjecture. We prove the conjecture for ▫$k = 1, 2$▫ and prove that ▫$alpha_k(G times H) ge alpha_k(G)|V(H)| + alpha_k(H)|V(G)| - alpha_k(G) alpha_k(H)$▫ holds for any ▫$k$▫. |
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| Keywords: | matematika, teorija grafov, neodvisnostno število, kartezični produkt grafov, mathematics, graph theory, independence number, Cartesian product of graphs |
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| Year of publishing: | 2011 |
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| Number of pages: | str. 1377-1383 |
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| Numbering: | Vol. 32, no. 8 |
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| PID: | 20.500.12556/DKUM-51910  |
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| UDC: | 519.17 |
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| ISSN on article: | 0195-6698 |
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| COBISS.SI-ID: | 16079705 |
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| NUK URN: | URN:SI:UM:DK:NWLVHO8H |
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| Publication date in DKUM: | 10.07.2015 |
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| Views: | 1767 |
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| Downloads: | 74 |
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| Metadata: |    |
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| Categories: | Misc.
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