2010 Fiscal Year Final Research Report
Construction of single-path designs and 2-path designs
| Project/Area Number |
19540142
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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, 経営情報学部, 教授 (00136631)
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| Project Period (FY) |
2007 – 2010
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| Keywords | Dudeney集合 / 2パスデザイン |
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| Research Abstract |
A set of Hamilton cycles in a graph is called a Dudeney set if every 2-path lies on exactly one of the cycles. It has been conjectured that there is a Dudeney set in the complete graph of order n for all n. It is known that there exists a Dudeney set when n is even, but the conjecture is still unsettled when n is odd.
In this reseach, we defined a black 1-factor, and proved that if there exists a black 1-factor then we can construct a Dudeney set. We also proved that if there is a black 1-factor of order p+1, then 2 is a quadratic residue modulo p. Using this result, we obtained some new Dudeney sets.
We studied the existence problem of a Dudeney set in the complete bipartite graph of order 2n and we proved the following theorem. There exists a Dudeney set in the complete bipartite graph of order 2n when n=0,1,3 (mod 4), and there exists a double Dudeney set when n=2 (mod 4). When n=2 (mod 4), the existence problem of a Dudeney set remains open.
We studied the problem on a Dudeney set of 4-cycles, and we proved that there exists a Dudeney set of 4-cycles if and only if one of the following holds: (i) n is even, or (ii) n is odd and λis even.
We have studied graceful labelings of trees and have found new graceful labelings of some trees. We also obtained some results on construction of some combinatorial designs.
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