Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs
In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvaria...
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MDPI AG
2021-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/5/484 |
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| author | Anita Keszler Zsolt Tuza |
| author_facet | Anita Keszler Zsolt Tuza |
| author_sort | Anita Keszler |
| collection | DOAJ |
| description | In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">C</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> of order <i>v</i> is a collection of <i>r</i>-uniform <i>k</i>-cycles on a <i>v</i>-element vertex set, such that each <i>r</i>-element subset is an edge in precisely one of those <i>k</i>-cycles. We present cyclic hypercycle systems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">C</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> of orders <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>=</mo><mn>25</mn><mo>,</mo><mn>26</mn><mo>,</mo><mn>31</mn><mo>,</mo><mn>35</mn><mo>,</mo><mn>37</mn><mo>,</mo><mn>41</mn><mo>,</mo><mn>46</mn><mo>,</mo><mn>47</mn><mo>,</mo><mn>55</mn><mo>,</mo><mn>56</mn></mrow></semantics></math></inline-formula>, a highly symmetric construction for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>=</mo><mn>40</mn></mrow></semantics></math></inline-formula>, and cyclic 2-split constructions of orders <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>32</mn><mo>,</mo><mn>40</mn><mo>,</mo><mn>50</mn><mo>,</mo><mn>52</mn></mrow></semantics></math></inline-formula>. As a consequence, all orders <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>≤</mo><mn>60</mn></mrow></semantics></math></inline-formula> permitted by the divisibility conditions admit a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">C</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> system. New recursive constructions are also introduced. |
| first_indexed | 2024-03-09T00:29:06Z |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T00:29:06Z |
| publishDate | 2021-02-01 |
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| series | Mathematics |
| spelling | doaj.art-1e151b016a8f434b9773156913e367eb2023-12-11T18:36:25ZengMDPI AGMathematics2227-73902021-02-019548410.3390/math9050484Hypercycle Systems of 5-Cycles in Complete 3-Uniform HypergraphsAnita Keszler0Zsolt Tuza1Machine Perception Laboratory, Institute for Computer Science and Control (SZTAKI), Kende u. 13-17, 1111 Budapest, HungaryAlfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15, 1053 Budapest, HungaryIn this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">C</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> of order <i>v</i> is a collection of <i>r</i>-uniform <i>k</i>-cycles on a <i>v</i>-element vertex set, such that each <i>r</i>-element subset is an edge in precisely one of those <i>k</i>-cycles. We present cyclic hypercycle systems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">C</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> of orders <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>=</mo><mn>25</mn><mo>,</mo><mn>26</mn><mo>,</mo><mn>31</mn><mo>,</mo><mn>35</mn><mo>,</mo><mn>37</mn><mo>,</mo><mn>41</mn><mo>,</mo><mn>46</mn><mo>,</mo><mn>47</mn><mo>,</mo><mn>55</mn><mo>,</mo><mn>56</mn></mrow></semantics></math></inline-formula>, a highly symmetric construction for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>=</mo><mn>40</mn></mrow></semantics></math></inline-formula>, and cyclic 2-split constructions of orders <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>32</mn><mo>,</mo><mn>40</mn><mo>,</mo><mn>50</mn><mo>,</mo><mn>52</mn></mrow></semantics></math></inline-formula>. As a consequence, all orders <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>≤</mo><mn>60</mn></mrow></semantics></math></inline-formula> permitted by the divisibility conditions admit a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">C</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> system. New recursive constructions are also introduced.https://www.mdpi.com/2227-7390/9/5/484hypergraphhypercycle system3-uniform 5-cycleedge decompositionSteiner system |
| spellingShingle | Anita Keszler Zsolt Tuza Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs Mathematics hypergraph hypercycle system 3-uniform 5-cycle edge decomposition Steiner system |
| title | Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs |
| title_full | Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs |
| title_fullStr | Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs |
| title_full_unstemmed | Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs |
| title_short | Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs |
| title_sort | hypercycle systems of 5 cycles in complete 3 uniform hypergraphs |
| topic | hypergraph hypercycle system 3-uniform 5-cycle edge decomposition Steiner system |
| url | https://www.mdpi.com/2227-7390/9/5/484 |
| work_keys_str_mv | AT anitakeszler hypercyclesystemsof5cyclesincomplete3uniformhypergraphs AT zsolttuza hypercyclesystemsof5cyclesincomplete3uniformhypergraphs |