On Hamilton Cycle Decomposition of 6-regular Circulant Graphs

Given two 2-regular graphs F 1 and F 2 , both of order n, the Hamilton-Waterloo Problem for F 1 and F 2 asks for a factorisation of the complete graph K n into α 1 copies of F 1 , α 2 copies of F 2 , and a 1-factor if n is even, for all non-negative integers α 1 and α 2 satisfying α 1 + α 2 = n−1 2 . We settle the Hamilton-Waterloo problem for all bipartite 2-regular graphs F 1 and F 2 where F 1 can be obtained from F 2 by replacing each cycle with a bipartite 2-regular graph of the same order.

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“…The first was proved by Bermond et al [7], and the second by the third author of the current paper [17]. Both…”

Section: Notation Definitions and Existing Resultsmentioning

confidence: 72%

On the Hamilton‐Waterloo Problem for Bipartite 2‐Factors

Bryant

Danziger

Dean

2012

J of Combinatorial Designs

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“…The first was proved by Bermond et al [6], and the second by Dean [13]. Both results address the open question of whether every connected Cayley graph of even degree on a finite abelian group has a Hamilton cycle decomposition [3].…”

Section: Preliminary Results and Notationmentioning

confidence: 83%

“…Theorem 5 (Dean [13]). Every 6-regular Cayley graph on a cyclic group which has a generator of the group in its connection set has a Hamilton cycle decomposition.…”

Section: Preliminary Results and Notationmentioning

confidence: 99%

On bipartite 2-factorizations of kn − I and the Oberwolfach problem

2010

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Lemma 12. There exist decompositions of the following types: 8+[4, 9+[4, 7 which we denote by L 2 . J 12 → [6, 6+[6, 7+[6, 5 which we denote by L 3 . J 16 → Most of the decompositions J |V(F)| → F that we need can be obtained from the decompositions given in Lemma 16 by using the ⊕ operation, but we also need a few other small decompositions and these are given in the following lemma. Lemma 17. The following decompositions exist: J 8 → [8], J 12 → [12], J 8 → [4, 4], J 12 → [4, 8], J 16 → [4, 12], J 12 → [6, 6], J 16 → [6, 10], J 20 → [10, 10], J 12 → [4, 4, 4], J 16 → [4, 6, 6], J 20 → [4, 6, 10], J 24 → [4, 10, 10]. Proof. The decomposition J 8 → [8] is given

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“…The first was proved by Bermond et al [18], and the second by Dean [53]. Both results address the open question of whether every connected Cayley graph of even degree on a finite abelian group has a Hamilton decomposition [4].…”

Section: Notation and Preliminariesmentioning

confidence: 78%

“…Since 3t + 4q + h = n−1 2 − 6, and since we have already dealt with the cases where (n, t) ∈ {(27, 1), (28, 1), (52,5), (70,8), (72,8)}, this leaves us with the cases where (n, t, h) in {(27, 1, 4), (29,1,5), (30,1,5), (51,5,4), (53,5,5), (54,5,5), (69,8,4), (71,8,5)}.…”

Section: T=1mentioning

confidence: 99%

Computational Graph Theory

Pettersson¹

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This thesis involves the application of computational techniques to various problems in graph theory and low dimensional topology. The first two chapters of this thesis focus on problems in graph theory itself; in particular on graph decomposition problems. The last three chapters look at applications of graph theory to combinatorial topology, focusing on the exhaustive generation of certain families of 3-manifold triangulations.

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On Hamilton Cycle Decomposition of 6-regular Circulant Graphs

Cited by 15 publications

References 4 publications

On the Hamilton‐Waterloo Problem for Bipartite 2‐Factors

On the Hamilton‐Waterloo Problem for Bipartite 2‐Factors

On bipartite 2-factorizations of kn − I and the Oberwolfach problem

Computational Graph Theory

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