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.
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Received: October, 2001 Final version received: September 4, 2002
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ID="*" This research was supported in part by Grant-in-Aid for Scientific Research (C) Japan
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Kobayashi, M., Nakamura, G. Resolvable Coverings of 2-Paths by Cycles. Graphs Comb 18, 739–744 (2002). https://doi.org/10.1007/s003730200059
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DOI: https://doi.org/10.1007/s003730200059