Twofold 2-perfect bowtie systems with an extra property
A bowtie is a closed trail whose graph consists of two 3-cycles with exactly one vertex in common. A 2-fold bowtie system of order n is an edge-disjoint decomposition of 2 K n into bowties. A 2-fold bowtie system is said to be 2-perfect provided that every pair of distinct vertices is joined by two...
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| Published in | Aequationes mathematicae Vol. 82; no. 1-2; pp. 143 - 153 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Basel
SP Birkhäuser Verlag Basel
01.09.2011
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0001-9054 1420-8903 |
| DOI | 10.1007/s00010-011-0075-0 |
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| Abstract | A bowtie is a closed trail whose graph consists of two 3-cycles with exactly one vertex in common. A 2-fold bowtie system of order
n
is an edge-disjoint decomposition of 2
K
n
into bowties. A 2-fold bowtie system is said to be 2-perfect provided that every pair of distinct vertices is joined by
two
paths of length 2. It is said to be
extra
provided these two paths always have distinct midpoints. The extra property guarantees that the two paths
x
,
a
,
y
and
x
,
b
,
y
between every pair of vertices form a 4-cycle (
x
,
a
,
y
,
b
), and that the collection of all such 4-cycles is a
four
-fold 4-cycle system. We show that the spectrum for extra 2-perfect 2-fold bowtie systems is precisely the set of all
n
≡ 0 or 1 (mod 3),
. Additionally, with an obvious definition, we show that the spectrum for extra 2-perfect 2-fold maximum packings of 2
K
n
with bowties is precisely the set of all
n
≡ 2 (mod 3),
. |
|---|---|
| AbstractList | A bowtie is a closed trail whose graph consists of two 3-cycles with exactly one vertex in common. A 2-fold bowtie system of order n is an edge-disjoint decomposition of 2K sub( )ninto bowties. A 2-fold bowtie system is said to be 2-perfect provided that every pair of distinct vertices is joined by two paths of length 2. It is said to be extra provided these two paths always have distinct midpoints. The extra property guarantees that the two paths x, a, y and x, b, y between every pair of vertices form a 4-cycle (x, a, y, b), and that the collection of all such 4-cycles is a four-fold 4-cycle system. We show that the spectrum for extra 2-perfect 2-fold bowtie systems is precisely the set of all n 0 or 1 (mod 3), n >= 6 . Additionally, with an obvious definition, we show that the spectrum for extra 2-perfect 2-fold maximum packings of 2K sub( )nwith bowties is precisely the set of all n 2 (mod 3), n >= 8 . A bowtie is a closed trail whose graph consists of two 3-cycles with exactly one vertex in common. A 2-fold bowtie system of order n is an edge-disjoint decomposition of 2K ^sub n^ into bowties. A 2-fold bowtie system is said to be 2-perfect provided that every pair of distinct vertices is joined by two paths of length 2. It is said to be extra provided these two paths always have distinct midpoints. The extra property guarantees that the two paths x, a, y and x, b, y between every pair of vertices form a 4-cycle (x, a, y, b), and that the collection of all such 4-cycles is a four-fold 4-cycle system. We show that the spectrum for extra 2-perfect 2-fold bowtie systems is precisely the set of all n 0 or 1 (mod 3), n 6 . Additionally, with an obvious definition, we show that the spectrum for extra 2-perfect 2-fold maximum packings of 2K ^sub n^ with bowties is precisely the set of all n 2 (mod 3), n 8 .[PUBLICATION ABSTRACT] A bowtie is a closed trail whose graph consists of two 3-cycles with exactly one vertex in common. A 2-fold bowtie system of order n is an edge-disjoint decomposition of 2 K n into bowties. A 2-fold bowtie system is said to be 2-perfect provided that every pair of distinct vertices is joined by two paths of length 2. It is said to be extra provided these two paths always have distinct midpoints. The extra property guarantees that the two paths x , a , y and x , b , y between every pair of vertices form a 4-cycle ( x , a , y , b ), and that the collection of all such 4-cycles is a four -fold 4-cycle system. We show that the spectrum for extra 2-perfect 2-fold bowtie systems is precisely the set of all n ≡ 0 or 1 (mod 3), . Additionally, with an obvious definition, we show that the spectrum for extra 2-perfect 2-fold maximum packings of 2 K n with bowties is precisely the set of all n ≡ 2 (mod 3), . |
| Author | Meszka, Mariusz Billington, Elizabeth J. Lindner, C. C. |
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| Keywords | bowtie Complete graph decomposition 05C38 2-perfect 05B30 |
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| References | Billington, Lindner (CR2) 1994; 135 Vanstone, Stinson, Schellenberg, Rosa, Rees, Colbourn, Carter, Carter (CR7) 1993; 3 CR3 CR6 Colbourn, Dinitz (CR4) 2006 Adams (CR1) 1993; 44 Colbourn, Rosa (CR5) 1999 75_CR3 C.J. Colbourn (75_CR5) 1999 S.A. Vanstone (75_CR7) 1993; 3 75_CR6 P. Adams (75_CR1) 1993; 44 (75_CR4) 2006 E.J. Billington (75_CR2) 1994; 135 |
| References_xml | – year: 1999 ident: CR5 publication-title: Triple Systems – ident: CR6 – volume: 44 start-page: 243 year: 1993 end-page: 253 ident: CR1 article-title: Lambda-fold 2-perfect bowties publication-title: Utilitas Mathematica – volume: 3 start-page: 305 issue: 3 year: 1993 end-page: 319 ident: CR7 article-title: Hanani triple systems publication-title: Israel J. Math. doi: 10.1007/BF02784058 – volume: 135 start-page: 61 year: 1994 end-page: 68 ident: CR2 article-title: The spectrum for 2-perfect bowtie systems publication-title: Discrete Math. doi: 10.1016/0012-365X(93)E0008-R – ident: CR3 – year: 2006 ident: CR4 publication-title: The Handbook of Combinatorial Designs – ident: 75_CR3 – volume-title: Triple Systems year: 1999 ident: 75_CR5 doi: 10.1093/oso/9780198535768.001.0001 – volume: 44 start-page: 243 year: 1993 ident: 75_CR1 publication-title: Utilitas Mathematica – volume: 3 start-page: 305 issue: 3 year: 1993 ident: 75_CR7 publication-title: Israel J. Math. doi: 10.1007/BF02784058 – volume-title: The Handbook of Combinatorial Designs year: 2006 ident: 75_CR4 – ident: 75_CR6 – volume: 135 start-page: 61 year: 1994 ident: 75_CR2 publication-title: Discrete Math. doi: 10.1016/0012-365X(93)E0008-R |
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| Snippet | A bowtie is a closed trail whose graph consists of two 3-cycles with exactly one vertex in common. A 2-fold bowtie system of order
n
is an edge-disjoint... A bowtie is a closed trail whose graph consists of two 3-cycles with exactly one vertex in common. A 2-fold bowtie system of order n is an edge-disjoint... |
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| SubjectTerms | Analysis Collection Combinatorics Decomposition Graph algorithms Graphs Mathematics Mathematics and Statistics |
| Title | Twofold 2-perfect bowtie systems with an extra property |
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