One-factorizations of complete multigraphs arising from maximal (k;n)-arcs in PG(2,2h)

Sonnino, Angelo

doi:10.1016/s0012-365x(00)00337-x

Cited by 8 publications

(11 citation statements)

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“…From the literature on indecomposable 1-factorizations we note that the existence of such 1factorizations in 8 , with > 4, is not known. Unfortunately, our constructions do not allow us to find results for 2 = 8.…”

Section: Final Remark Smentioning

confidence: 99%

“…Simple and indecomposable 1‐factorizations of

false(n - d false) K_{2 n}

, with

d \geq 2

n - d \geq 5

and

gcd (n, d) = 1

, are constructed in . Other values of λ and n for which the existence of a simple and indecomposable 1‐factorization of

λ K_{2 n}

is known are the following:

2 n = q^{2} + 1

λ = q - 1

, where q is an odd prime power (see );

2 n = 2^{h} + 2

λ = 2

(see );

2 n = q^{2} + 1

λ = q + 1

, where q is an odd prime power (see );

2 n = q^{2}

λ = q

, where q is an even prime power (see ).…”

Section: Introductionmentioning

confidence: 99%

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Indecomposable 1‐factorizations of the complete multigraph for every

Bonvicini

Rinaldi

2017

J of Combinatorial Designs

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A 1-factorization of the complete multigraph 2 is said to be indecomposable if it cannot be represented as the union of 1-factorizations of 0 2 and ( − 0 ) 2 , where 0 < . It is said to be simple if no 1-factor is repeated. For every ≥ 9 and for every ( − 2)∕3 ≤ ≤ 2 , we construct an indecomposable 1-factorization of 2 , which is not simple. These 1-factorizations provide simple and indecomposable 1-factorizations of 2 for every ≥ 18 and 2 ≤ ≤ 2⌊ ∕2⌋ − 1. We also give a generalization of a result by Colbourn et al., which provides a simple and indecomposable 1-factorization of 2 , where 2 = + 1, = ( − 1)∕2, prime.

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Section: Final Remark Smentioning

confidence: 99%

“…Simple and indecomposable 1‐factorizations of

false(n - d false) K_{2 n}

, with

d \geq 2

n - d \geq 5

and

gcd (n, d) = 1

, are constructed in . Other values of λ and n for which the existence of a simple and indecomposable 1‐factorization of

λ K_{2 n}

is known are the following:

2 n = q^{2} + 1

λ = q - 1

, where q is an odd prime power (see );

2 n = 2^{h} + 2

λ = 2

(see );

2 n = q^{2} + 1

λ = q + 1

, where q is an odd prime power (see );

2 n = q^{2}

λ = q

, where q is an even prime power (see ).…”

Section: Introductionmentioning

confidence: 99%

Indecomposable 1‐factorizations of the complete multigraph for every

Bonvicini

Rinaldi

2017

J of Combinatorial Designs

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“…Kiss, G. Korchmáros and the authors used geometry in order to set up a procedure for the construction of (possibly new) families of one-factorisations based on nice geometric objects. The idea of using geometry for the construction of factorisations dates back to 2001 when it was first exploited on multigraphs; see for instance [1,9,11,15,19]. In this paper, we construct one-factorisations from ovals in a Desarguesian projective plane of odd square order, the order of which is either 9 2 , or the square of a prime of the form 2 d + 1.…”

Section: Introductionmentioning

confidence: 99%

One-Factorisations of Complete Graphs Constructed in Desarguesian Planes of Certain Odd Square Orders

Pace

Sonnino

2020

Electron. J. Combin.

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“…Only a few conditions on the parameters are known: if IOF(2n, λ) exists, then λ < 1 · 3 · ... · (2n − 3) [4]; each IOF(2n, λ) can be embedded in a simple IOF(2s, λ), provided that λ < 2n < s [16]. Six infinite classes of indecomposable one-factorizations have been constructed so far, namely a simple IOF(2n, n − 1) when 2n − 1 is a prime [16], IOF(2(λ + p), λ) where λ > 2 and p is the smallest prime wich does not divide λ [3] (an improvement of this result can be found in [15]), a simple IOF(2 h + 2, 2) where h is a positive integer [27], IOF(q 2 + 1, q − 1) where q is an odd prime number [26], a simple IOF(q 2 + 1, q + 1) for any odd prime power q [25], and a simple IOF(q 2 , q) for any even prime power q [25]. Most of these constructions arise from finite geometry.…”

Section: Introductionmentioning

confidence: 99%