The extended metamorphosis of a complete bipartite design into a cycle system
A K t, t -design of order n is an edge-disjoint decomposition of K n into copies of K t, t . When t is odd, an extended metamorphosis of a K t, t -design of order n into a 2 t-cycle system of order n is obtained by taking ( t−1)/2 edge-disjoint cycles of length 2 t from each K t, t block, and rearra...
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Published in | Discrete mathematics Vol. 284; no. 1; pp. 63 - 70 |
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Main Author | |
Format | Journal Article |
Language | English |
Published |
Amsterdam
Elsevier B.V
06.07.2004
Elsevier |
Subjects | |
Online Access | Get full text |
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Summary: | A
K
t,
t
-design of order
n is an edge-disjoint decomposition of
K
n
into copies of
K
t,
t
. When
t is odd, an extended metamorphosis of a
K
t,
t
-design of order
n into a 2
t-cycle system of order
n is obtained by taking (
t−1)/2 edge-disjoint cycles of length 2
t from each
K
t,
t
block, and rearranging all the remaining 1-factors in each
K
t,
t
block into further 2
t-cycles. The ‘extended’ refers to the fact that as many subgraphs isomorphic to a 2
t-cycle as possible are removed from each
K
t,
t
block, rather than merely one subgraph.
In this paper an extended metamorphosis of a
K
t,
t
-design of order congruent to
1
(
mod
4t
2)
into a 2
t-cycle system of the same order is given for all odd
t>3. A metamorphosis of a 2-fold
K
t,
t
-design of any order congruent to
1
(
mod
t
2)
into a 2
t-cycle system of the same order is also given, for all odd
t>3. (The case
t=3 appeared in Ars Combin. 64 (2002) 65–80.)
When
t is even, the graph
K
t,
t
is easily seen to contain
t/2 edge-disjoint cycles of length 2
t, and so the metamorphosis in that case is straightforward. |
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ISSN: | 0012-365X 1872-681X |
DOI: | 10.1016/j.disc.2003.11.025 |