Resolvable Coverings of 2-Paths by Cycles |
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Authors: | Midori Kobayashi Gisaku Nakamura |
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Institution: | (1) School of Administration and Informatics, University of Shizuoka, Shizuoka 422-8526, Japan. e-mail: midori@u-shizuoka-ken.ac.jp, JP;(2) Tokai University, Shibuya-ku, Tokyo 151-0063, Japan, JP |
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Abstract: | Let K
n
be the complete graph on n vertices. A C(n,k,λ) design is a multiset of k-cycles in K
n
in which each 2-path (path of length 2) of K
n
occurs exactly λ times. A C(lk,k,1) design is resolvable if its k-cycles can be partitioned into classes so that every vertex appears exactly once in each class.
A C(n,n,1) design gives a solution of Dudeney's round table problem. It is known that there exists a C(n,n,1) design when n is even and there exists a C(n,n,2) design when n is odd. In general the problem of constructing a C(n,n,1) design is still open when n is odd. Necessary and sufficient conditions for the existence of C(n,k,λ) designs and resolvable C(lk,k,1) designs are known when k=3,4.
In this paper, we construct a resolvable C(n,k,1) design when n=p
e
+1 ( p is a prime number and e≥1) and k is any divisor of n with k≠1,2.
Received: October, 2001 Final version received: September 4, 2002
RID="*"
ID="*" This research was supported in part by Grant-in-Aid for Scientific Research (C) Japan |
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Keywords: | |
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