Curran, S.J., Gallian, J.A.: Hamiltonian cycles and paths in Cayley graphs and digraphs---a survey. Discrete Math. 156 (1996) 1--18  Home/Search   Document Not in Database   Summary   Related Articles   Check

....all unordered pairs ffg; gsg : s Sg. This is obviously a regular graph of degree jS [ S #1 j#2jSj. The question of Hamiltonicity of Cayley graphs has drawn quite an amount of attention of many researchers over the years. It is enough to mention that a survey article by Curran and Gallian [12] on Hamiltonicity of Cayley graphs has eighty nine references. Much of the focus of the research has been centered around the following conjecture. Conjecture 4.1. Every connected Cayley graph with more than 2 vertices is Hamiltonian. So far only special cases of the above conjecture have been ....

S. J. Curran and J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs---a survey, Discrete Math 156 (1996), 1--18.

....of elements from S that form the arcs. Moreover, since the same collection of arcs give a hamiltonian path from any starting vertex, we only need to give the arcs of the path. The question which Cayley graphs and digraphs contain a hamiltonian path ( or cycle ) has a long history. See, e.g. [3, 16] for a survey. For our purposes we only need the following special, easy result. 6.2 Lemma Let G be a finite abelian group and S G, 0 62 S. Then D(G;S) contains a hamiltonian path if and only if S generates G. Proof Since D = D(G;S) is connected if and only if S generates the group G, it is ....

S.J. Curran and J.A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs -- A survey. Discrete Math. 156 (1996) 1--18.

....have been dealing with sufficient conditions for the existence of a decomposition of a graph into hamiltonian cycles. But still, the current status of the matter lies, for the most part, in the sphere of problems and conjectures. A survey on hamiltonian decomposition of graphs is provided in [3, 4]. The complete graph Kn with odd (resp. even) number n of vertices is decomposable into hamiltonian cycles (resp. paths) The complete k partite graph K(n 1 ; n 2 ; Delta Delta Delta ; n k ) is hamiltonian decomposable if and only if n 1 = n 2 = Delta Delta Delta = n k . The following ....

S. J. Curran and J. A. Gallian, "Hamiltonian cycles and paths in Cayley graphs and digraphs - a survey," Discrete Mathematics 156, pp. 1-18, 1996.

....decomposition can be regarded as two directed hamiltonian cycles of opposite direction. When r is odd, it is not always true. This work was supported by the Korea Science and Engineering Foundation under grant no. 98 0102 07 01 3. A survey on hamiltonian decomposition of graphs is provided in[3, 4]. Much of the focus of research has been directed towards proving that some special cases of Cayley graphs over an abelian group are hamiltonian decomposable, such as the product of any number of cycles, m cubes, and recursive circulants[5, 9] But still, the current status of the matter lies, for ....

S. J. Curran and J. A. Gallian, "Hamiltonian cycles and paths in cayley graphs and digraphs - a survey," Discrete Mathematics 156, pp. 1-18, 1996.

....undirected, vertex transitive graph has a Hamilton path [Lov70] Results on Hamilton cycles are surveyed in [Als81] for vertex transitive graphs and in [Gou91] for general graphs. A survey of Hamilton cycles in Cayley graphs can be found in [WG84] and in the recent update of Curran and Gallian [CG96]. We focus here on a few recent questions which arose in the context of Gray codes. Suppose the group G is S n , the symmetric group of permutations of n symbols, and let 28 X be a generating set of S n . Then a Hamilton cycle in the Cayley graph C[G; X ] can be regarded as a Gray code for ....

....survey of Squire [Squ94a] In [Gol93] Goldberg considers generating combinatorial structures for which achieving even polynomial delay is hard. For surveys on related material, see [Als81] for long cycles in vertex transitive graphs, Gou91] for hamiltonian cycles, WG84] and the recent update [CG96] for Cayley graphs, and [Sed77] for permutations. Acknowledgements I am grateful to Herb Wilf for collecting and sharing such an intriguing array of Gray code problems. His work, as well as his enthusiasm, has been inspiring. I would also like to thank Frank Ruskey, my frequent co author, for ....

S. J. Curran and J. A. Gallian. Hamiltonian cycles and paths in Cayley graphs and digraphs - a survey, 1996.

....of a single generator or its inverse to its immediate predecessor This problem seems very difficult. For surveys of the general topic of Gray codes, for many more examples of such codes in a variety of combinatorial families, and for pointers to recent literature in the subject we suggest [1, 3, 6, 8]. 1.2. About this paper. In this paper, we study a numerical obstruction to being able to list in Gray code order a collection of subsets of f1; ng. We show that this obstruction grows exponentially in n for the collection of g blockfree subsets of f1; ng if and only if g 2. ....

S. J. Curran and J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs - a survey, Discrete Math. 156 (1996), 1-18.

.... Gamma 1) Hamilton cycles and a 1 factor when n is odd. This property is important in many applications. In [1] B. Alspach asked the following question: Is it the case that every connected Cayley graph X(G;H) on an abelian group G admits a Hamilton decomposition Some cases have been resolved [5]. B. Alspach proved that Hamilton decomposability of vertex transitive Cayley graphs of order 2p, where p is prime and p j 3(mod 4) J.C. Bermond, O. Favaron and M. Maheo were interested in Cayley graphs of degree 4 [4] D. Barth and A. Raspaud constructed two edge disjoint Hamiltonian cyles in ....

Stephen J. Curran, Joseph A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs -- a survey, 1994, preprint.

....of these graphs, only the Petersen graph is not hamiltonian. Key words: graph, vertex transitive, Hamilton cycle, commutator subgroup 1 Introduction Considerable attention has been devoted to the problem of determining whether or not a connected, vertex transitive graph X has a Hamilton cycle [1] [8], 14] A graph X is vertex transitive if some group G of automorphisms of X Preprint submitted to Discrete Mathematics 5 December acts transitively on V (X) If G is abelian, then it is easy to see that X has a Hamilton cycle. Thus it is natural to try to prove the same conclusion when G is ....

S.J. Curran and J.A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs --- a survey, Discrete Math. 156 (1996) 1--18.

....for every proper subset A # of A, the subdigraph Cay(Z n ; A # ) is not connected, then Cay(Z n ; A) has a hamiltonian circuit. As mentioned above, circulant digraphs are Cayley digraphs on cyclic groups. Thus, this paper is related to the literature on hamiltonian circuits in Cayley digraphs [1] [3], 6] Indeed, Rankin s Theorem (1.2) was proved for 3 2 generated Cayley digraphs on any abelian group, not just on cyclic groups (and even some Cayley digraphs on nonabelian groups) Similarly, Theorem 1.3 and Conjecture 1.5 are only special cases of statements for all abelian groups. A basic ....

S. J. Curran and J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs --- a survey, Discrete Math. 156 (1996) 1--18.

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Curran, S.J., Gallian, J.A.: Hamiltonian cycles and paths in Cayley graphs and digraphs---a survey. Discrete Math. 156 (1996) 1--18

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S. J. Curran and J. A. Gallian. Hamiltonian cycles and paths in Cayley graphs and digraphs---a survey. Discrete Math., 156(1-3):1--18, 1996.

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S. J. Curran, J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs| a survey, Discrete Math. 156 (1996), 1-18.

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S. J. Curran and J. A. Gallian, \Hamiltonian cycles and paths in Cayley graphs and digraphs - a survey," Discrete Mathematics 156 , pp. 1-18, 1996.

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