Decomposition of the complete graph plus a 1‐factor into cycles of equal length

Šajna, Mateja

doi:10.1002/jcd.10019

Cited by 9 publications

(4 citation statements)

References 8 publications

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“…They claimed (without proof) that the conjecture holds for | | ≤ A 5 and verified it by computer for ≤ n 16. They stated that Alspach was motivated by the existence of cycle decompositions of complete graphs and complete graphs plus or minus a 1-factor [5,22,23], and of directed cycle decompositions of complete symmetric digraphs [6].…”

Section: Introductionmentioning

confidence: 99%

Distinct partial sums in cyclic groups: polynomial method and constructive approaches

Hicks¹,

Ollis

Schmitt

2019

J of Combinatorial Designs

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Let ( G , + ) be an abelian group and consider a subset A ⊆ G with ∣ A ∣ = k. Given an ordering ( a 1 , … , a k ) of the elements of A, define its partial sums by s 0 = 0 and s j = ∑ i = 1 j a i for 1 ≤ j ≤ k. We consider the following conjecture of Alspach: for any cyclic group Z n and any subset A ⊆ Z n ⧹ { 0 } with s k ≠ 0, it is possible to find an ordering of the elements of A such that no two of its partial sums s i and s j are equal for 0 ≤ i < j ≤ k. We show that Alspach’s Conjecture holds for prime n when k ≥ n − 3 and when k ≤ 10. The former result is by direct construction, the latter is nonconstructive and uses the polynomial method. We also use the polynomial method to show that for prime n a sequence of length k having distinct partial sums exists in any subset of Z n ⧹ { 0 } of size at least 2 k − 8 k in all but at most a bounded number of cases.

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Section: Introductionmentioning

confidence: 99%

Distinct partial sums in cyclic groups: polynomial method and constructive approaches

Hicks¹,

Ollis

Schmitt

2019

J of Combinatorial Designs

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“…They claimed (without proof) that the conjecture holds for |A| ≤ 5 and verified it by computer for n ≤ 16. They stated that Alspach was motivated by the existence of cycle decompositions of complete graphs and complete graphs plus or minus a 1-factor (see [5], [22], and [23]), and of directed cycle decompositions of complete symmetric digraphs, [6].…”

Section: Introductionmentioning

confidence: 99%

Distinct Partial Sums in Cyclic Groups: Polynomial Method and Constructive Approaches

Hicks¹,

Ollis²,

Schmitt³

2018

Preprint

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Let (G, +) be an abelian group and consider a subset A ⊆ G with |A| = k. Given an ordering (a 1 , . . . , a k ) of the elements of A, define its partial sums by s 0 = 0 and s j = j i=1 a i for 1 ≤ j ≤ k. We consider the following conjecture of Alspach: For any cyclic group Z n and any subset A ⊆ Z n \ {0} with s k = 0, it is possible to find an ordering of the elements of A such that no two of its partial sums s i and s j are equal for 0 ≤ i < j ≤ k. We show that Alspach's Conjecture holds for prime n when k ≥ n − 3 and when k ≤ 10. The former result is by direct construction, the latter is non-constructive and uses the polynomial method. We also use the polynomial method to show that for prime n a sequence of length k having distinct partial sums exists in any subset of Z n \ {0} of size at least 2k − √ 8k in all but at most a bounded number of cases.

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“…Obvious necessary condition for the existence of a -cycle decomposition of a simple connected graph is that has at least vertices (or trivially, just one vertex), the degree of every vertex in is even, and the total number of edges in is a multiple of the cycle length . These conditions have been shown to be sufficient in the case that is the complete graph , the complete graph minus a 1-factor − [1,2], and the complete graph plus a 1-factor + [3]. The study of cycle decomposition of * was initiated by Hoffman et al [4].…”

Section: Introductionmentioning

confidence: 99%

Sunlet Decomposition of Certain Equipartite Graphs

Akwu

Ajayi

2013

International Journal of Combinatorics

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Let L2n stand for the sunlet graph which is a graph that consists of a cycle and an edge terminating in a vertex of degree one attached to each vertex of cycle Cn. The necessary condition for the equipartite graph Kn+I*K̅m to be decomposed into L2n for n≥2 is that the order of L2n must divide n2m2/2, the order of Kn+I*K̅m. In this work, we show that this condition is sufficient for the decomposition. The proofs are constructive using graph theory techniques.

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Decomposition of the complete graph plus a 1‐factor into cycles of equal length

Abstract: We determine the necessary and sufficient conditions for the existence of a decomposition of the complete graph of even order with a 1-factor added into cycles of equal length. #

Cited by 9 publications

References 8 publications

Distinct partial sums in cyclic groups: polynomial method and constructive approaches

Distinct partial sums in cyclic groups: polynomial method and constructive approaches

Distinct Partial Sums in Cyclic Groups: Polynomial Method and Constructive Approaches

Sunlet Decomposition of Certain Equipartite Graphs

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