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Hamiltonian cycles and paths in Cayley graphs and digraphs — A survey

Stephen J. Curran, Joseph A. Gallian
1996 Discrete Mathematics  
Since the 1984 survey of results on hamiltonian cycles and paths in Cayley graphs by Witte and Gallian, many advances have been made.  ...  Moreover, Babai has shown that all graphs can be realized as an induced subgraph of a Cayley graph of any sufficiently large group.  ...  We wish to thank David Witte, Douglas Jungreis and the referees for their comments and suggestions.  ... 
doi:10.1016/0012-365x(95)00072-5 fatcat:wlr3et6cdvhediytnvdz2tnetm

A survey: Hamiltonian cycles in Cayley graphs

David Witte, Joseph A. Gallian
1984 Discrete Mathematics  
It has been conjectured there is a hamiltonian cycle in every Cayley graph. Interest in this and other closely related questions has grown in the past few years. We su~-;ey the ~PCU!!  ...  C, techniques and open problems in the field.  ...  Parsons and others). There is a hamiltonian cycle in every Cayley graph. We are nowhere near a prooaf of this conjecture.  ... 
doi:10.1016/0012-365x(84)90010-4 fatcat:vgd2fxdwgnhkvnorhzlkbh2g6m

Updating the hamiltonian problem—A survey

Ronald J. Gould
1991 Journal of Graph Theory  
This is intended as a survey article covering recent developments in the area of hamiltonian graphs, that is, graphs containing a spanning cycle.  ...  This article also contains some material on related topics such as traceable, hamiltonian-connected and pancyclic graphs and digraphs, as well as an extensive bibliography of papers in the area. 1.  ...  In 1981, Bermond and Thomassen [41] gave an outstanding survey of these and many other results on cycles in digraphs. I shall concentrate on subsequent work.  ... 
doi:10.1002/jgt.3190150204 fatcat:aagxz4hhj5cblm6w5couo6bkt4

A Survey of Combinatorial Gray Codes

Carla Savage
1997 SIAM Review  
In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems.  ...  This resulted in much recent activity in the area and most of the problems posed by Wilf are now solved.  ...  I would also like to thank Frank Ruskey, my frequent co-author, for many provoking discussions, and for his constant supply of interesting problems.  ... 
doi:10.1137/s0036144595295272 fatcat:dtl3y5ir5zhtnltw7g47eiwpeq

A survey on Knödel graphs

Guillaume Fertin, André Raspaud
2004 Discrete Applied Mathematics  
Kn odel graphs of even order n and degree 1 6 6 log 2 (n) , W ; n , are graphs which have been introduced some 25 years ago as the topology underlying a time optimal algorithm for gossiping among n nodes  ...  Since then, they have been widely studied as interconnection networks, mainly because of their good properties in terms of broadcasting and gossiping (Int.  ...  In that case, since we know that edges in dimensions 0 and 1 form a Hamiltonian cycle, then any edges in dimensions i and i + 1 (with 0 6 i 6 k − 3) also form a Hamiltonian cycle.  ... 
doi:10.1016/s0166-218x(03)00260-9 fatcat:m3q6sqez5betpgxjca3fvf7nai

A survey of repositories in graph theory [article]

Srinibas Swain, C. Paul Bonnington, Graham Farr, Kerri Morgan
2020 arXiv   pre-print
This survey reviews many of these graph repositories and summarises the scope and contents of each repository.  ...  Foster in the 1930s, many graph repositories have been created to support research in graph theory.  ...  Acknowledgement The authors would like to thank Charles Colbourn, Marston Conder, and Brendan McKay for their comments on earlier versions of this paper.  ... 
arXiv:2006.12741v1 fatcat:uxzb3xfahbesbdhgynn3y7kugu

Power graphs: A survey

Jemal Abawajy, Andrei Kelarev, Morshed Chowdhury
2013 Electronic Journal of Graph Theory and Applications  
This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature.  ...  Various conjectures due to other authors, questions and open problems are also included.  ...  The authors are grateful to Brian Curtin, Gholam Reza Pourgholi and Igor Shparlinski for comments and lists of typographical corrections to the manuscript.  ... 
doi:10.5614/ejgta.2013.1.2.6 fatcat:tgvssnqcfndilhnsxfaluevgqq

An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

Xueliang Li, Yuefang Sun
2017 Theory and Applications of Graphs  
There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013.  ...  Li and Sun: Rainbow connections of graphs -a dynamic survey Li and Sun: Rainbow connections of graphs -a dynamic survey Published by Digital Commons@Georgia Southern, 2017, Last updated 2017 σ 2 .  ...  Acknowledgement: The authors would like to thank the referees and the editor for their helpful suggestions and comments which helped to polish the presentation of the paper.  ... 
doi:10.20429/tag.2017.000103 fatcat:76awzxcw6ra3ldnibx5xtvtmmu

On Bondage Numbers of Graphs: A Survey with Some Comments

Jun-Ming Xu
2013 International Journal of Combinatorics  
The bondage number of a nonempty graph is the smallest number of edges whose removal from results in a graph with domination number greater than the domination number of .  ...  This paper gives a survey on the bondage number, including known results, conjectures, problems, and some comments, also selectively summarizes other types of bondage numbers.  ...  the references [128] and [81] , which resulted in this version.  ... 
doi:10.1155/2013/595210 fatcat:6zg7jz7i5jenzc2e2izfdy2txq

Connected Factors in Graphs ? a Survey

Mekkia Kouider, Preben Dahl Vestergaard
2005 Graphs and Combinatorics  
We observe that a graph G of order n has a Hamiltonian path if and only if the graph G ′ = G + K 1 , obtained from G by adding one new vertex joined to every vertex of G, has a Hamiltonian cycle.  ...  Some of the earliest results of graph theory on Hamiltonian cycles, spanning trees and walks fall naturally in that category.  ... 
doi:10.1007/s00373-004-0587-7 fatcat:jz5oy7wjtnbyxpr37rnwiaxhta

Page 721 of Mathematical Reviews Vol. , Issue 98B [page]

1998 Mathematical Reviews  
(Background on Hamilton cycles in Cayley graphs can be found in the surveys of the reviewer and J. A. Gallian [Discrete Math. 51 (1984), no. 3, 293- 304; MR 86a:05084] and S. C.  ...  In other words, there is a Hamilton cycle in the Cayley graph of PSL2(p) with respect to the generating set {A,X}.  ... 

Page 4556 of Mathematical Reviews Vol. , Issue 96h [page]

1996 Mathematical Reviews  
Summary: “We define a class I of 4-regular Cayley graphs on abelian groups and prove every element of I to be decomposable into two Hamiltonian cycles.  ...  infinite Hamiltonian paths.” 96h:05099 05C25 20827 Maller, Rognvaldur G.  ... 

Page 5297 of Mathematical Reviews Vol. , Issue 93j [page]

1993 Mathematical Reviews  
There exists a large number of possible Hamiltonian paths (cycles) in an n-dimensional hypercube Q,,.  ...  Summary: “Finding Hamiltonian paths (or cycles) in a hypercube has been the focus of many researchers.  ... 

Page 2 of Mathematical Reviews Vol. , Issue 98F [page]

1998 Mathematical Reviews  
Hamiltonian cycles and decompositions of Cayley digraphs of finite abelian groups.  ...  Summary: “A typical sufficient condition for the existence of a Hamiltonian cycle in a lexicographical product G[H j of two graphs G and H forces G to contain a Hamiltonian cycle or G to contain a Hamiltonian  ... 

Hamiltonian Cayley Digraphs on Direct Products of Dihedral Groups

Grant Andruchuk, Shonda Gosselin, Yizhe Zeng
2012 Open Journal of Discrete Mathematics  
We prove that a Cayley digraph on the direct product of dihedral groups D 2n × D 2m with outdegree two is Hamiltonian if and only if it is connected.  ...  A Hamilton cycle (path) in a digraph of with n vertices is a directed cycle (path) with n vertices. A digraph is said to be Hamiltonian if it has a Hamilton cycle.  ...  Since Cayley digraphs are vertextransitive, any cyclic shift of a Hamiltonian arc sequence of a Cayley digraph is also a Hamiltonian arc sequence of the digraph, and traversing a Hamiltonian arc sequence  ... 
doi:10.4236/ojdm.2012.23016 fatcat:ucw772n6anag7fxfxlsgem5wpa
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