Decomposing Various Graphs into Short Even-Length Cycles

Horsley, Daniel

doi:10.1007/s00026-012-0147-4

Cited by 12 publications

(57 citation statements)

References 19 publications

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“…This complements existing results for cycle decompositions of graphs such as the complete graph [1,3,10], complete bipartite graph [6,8] and complete multigraph [2].…”

supporting

confidence: 79%

“…This complements existing results for cycle decompositions of graphs such as the complete graph [1,3,10], complete bipartite graph [6,8] and complete multigraph [2]. [2] Generalisations of the Doyen-Wilson theorem 167 We obtain these cycle decomposition results by applying a cycle switching technique to modify cycle packings of K v − K u .…”

supporting

confidence: 66%

“…There has been much interest in the literature in generalising this result for m-cycle systems when m > 3. Although there are several partial results, including complete solutions for some small values of m and strong partial results for even m, this still remains an open problem [4,5,8,9].The main results of this thesis concern generalisations of the Doyen-Wilson theorem for odd m-cycle systems and cycle decompositions of the complete graph with a hole. The complete graph of order v with a hole of size u, K v − K u , is constructed from the complete graph of order v by removing the edges of a complete subgraph of order u (where v ≥ u).…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Generalisations of the Doyen–wilson Theorem

Hoyte

2017

Bull. Aust. Math. Soc.

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In 1973, Doyen and Wilson [7] famously solved the problem of when a 3-cycle system can be embedded in another 3-cycle system. There has been much interest in the literature in generalising this result for m-cycle systems when m > 3. Although there are several partial results, including complete solutions for some small values of m and strong partial results for even m, this still remains an open problem [4,5,8,9].The main results of this thesis concern generalisations of the Doyen-Wilson theorem for odd m-cycle systems and cycle decompositions of the complete graph with a hole. The complete graph of order v with a hole of size u, K v − K u , is constructed from the complete graph of order v by removing the edges of a complete subgraph of order u (where v ≥ u).For each odd m ≥ 3 we completely solve the problem of when an m-cycle system of order u can be embedded in an m-cycle system of order v, barring a finite number of possible exceptions. The problem is completely resolved in cases where u is large compared to m, where m is a prime power, or where m ≤ 15. In other cases, the only possible exceptions occur when v − u is small compared to m. This result is proved as a consequence of a more general result which gives necessary and sufficient conditions for the existence of an m-cycle decomposition of K v − K u in the case where u ≥ m − 2 and v − u ≥ m + 1 both hold.We prove that K v − K u can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, v − u ≥ 10, each cycle has length at most min (u, v − u), and the longest cycle is at most three times as long as the second longest. This complements existing results for cycle decompositions of graphs such as the complete graph [1,3,10], complete bipartite graph [6,8] and complete multigraph [2].

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“…This complements existing results for cycle decompositions of graphs such as the complete graph [1,3,10], complete bipartite graph [6,8] and complete multigraph [2].…”

supporting

confidence: 79%

supporting

confidence: 66%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Generalisations of the Doyen–wilson Theorem

Hoyte

2017

Bull. Aust. Math. Soc.

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“…Later on, the gap was completely filled by Ma et al [10]. Recently, Horsley [9] showed the existence of a cycle decomposition {C m 1 , C m 2 , . .…”

Section: Introductionmentioning

confidence: 97%

Decomposition of Complete Bipartite Graphs into Cycles of Distinct Even Lengths

Huang

Mishima

2015

Graphs and Combinatorics

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Small Embeddings of Partial Steiner Triple Systems

Horsley

2013

J of Combinatorial Designs

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It was proved in 2009 that any partial Steiner triple system of order u has an embedding of order v for each admissible . This result is best possible in the sense that, for each , there exists a partial Steiner triple system of order u that does not have an embedding of order v for any . Many partial Steiner triple systems do have embeddings of orders smaller than , but much less is known about when these embeddings exist. In this paper, we detail a method for constructing such embeddings. We use this method to show that each member of a wide class of partial Steiner triple systems has an embedding of order v for at least half (or nearly half) of the orders for which an embedding could exist. For some members of this class we are able to completely determine the set of all orders for which the member has an embedding.

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Decomposing Various Graphs into Short Even-Length Cycles

Cited by 12 publications

References 19 publications

Generalisations of the Doyen–wilson Theorem

Generalisations of the Doyen–wilson Theorem

Decomposition of Complete Bipartite Graphs into Cycles of Distinct Even Lengths

Small Embeddings of Partial Steiner Triple Systems

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