2005 Fiscal Year Final Research Report Summary
cycle design and path design
| Project/Area Number |
14540132
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | University of Shizuoka |
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Principal Investigator |
KOBAYASHI Midori University of Shizuoka, School of Administration and Informatics, professor, 経営情報学部, 教授 (00136631)
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| Project Period (FY) |
2002 – 2005
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| Keywords | Hamilton cycle / neighbor design / uniform covering / Dudeney set |
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| Research Abstract |
Let K_n be the complete graph on n vertices. A uniform covering of the 2-paths in K_n with l-paths [l-cycles] is a set S of l-paths [l-cycles] having the property that each 2-path in K_n lies in exactly one l-path [l-cycle] in S.
When l=n, the problem of constructing a uniform covering of the 2-paths with l-paths, is the famous Dudeney's round table problem which has not been solved yet.
For a given integer l, only the following cases of the problem of constructing a uniform covering of the 2-paths in K_n with l-paths or l-cycles have been solved : with 3-cycles, with 3-paths, with 4-cycles, with 4-paths, with n-cycles (Hamilton cycles) when n is even.
In this research, we solved the problem in the case of 5-paths, 6-paths and 6-cycles, that is, we settled the necessary and sufficient conditions on existing a uniform covering of 2-paths with 5-paths, 6-paths and 6-cycles in K_n. There are no known cases where the necessary conditions on n are not sufficient for the existence of a uniform covering of 2-paths in K_n.
We also constructed an antipodal Hamilton cycle decomposition, a double Dudeney set, resolvable designs and symmetric Hamilton cycle decompositions.
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