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Hamilton-decomposable graphs and digraphs of infinite valence

Dave Witte
1990 Discrete Mathematics  
We also consider directed graphs: we come close to a characterization of those digraphs that can be decomposed into infinitely many two-way-infinite Hamilton paths.  ...  A one-way-infinite Hamilton path in an infinite graph is a list 21i, u2, . . . of the vertices of the graph, in such a way that there is an edge joining q and Vi+* for each i.  ...  Acknowledgments This research was inspired by work of I. Jungreis [l] and K. Keating [2] . The research was begun at the University of Minnesota,  ... 
doi:10.1016/0012-365x(90)90275-m fatcat:jnkw73onsveshihaj6qe6wfpmm

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

1998 Mathematical Reviews  
The main purpose of the paper is to present a construction of an infinite family of finite one-regular graphs of valency 4.  ...  Jungreis and the reviewer's [Discrete Math. 143 (1995), no. 1-3, 1-30; MR 96f:20071] construction of infinite Hamilton paths in certain Cayley graphs of hyperbolic symmetry groups.  ... 

On Hamilton Decompositions of Infinite Circulant Graphs [article]

Darryn Bryant, Sarada Herke, Barbara Maenhaut, Bridget Webb
2017 arXiv   pre-print
We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the  ...  The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph).  ...  Acknowledgement: The authors acknowledge the support of the Australian Research Council (grants DP150100530, DP150100506, DP120100790 and DP130102987) and the EPSRC (grant EP/M016242/1).  ... 
arXiv:1701.08506v1 fatcat:fzucqqqiwbbm5ibdlc2vgqyy4m

Page 3572 of Mathematical Reviews Vol. , Issue 91G [page]

1991 Mathematical Reviews  
Yousef Alavi (1-WMI) 91g:05092 05C45 05C20 Witte, Dave (1-MSRI) Hamilton-decomposable graphs and digraphs of infinite valence. Discrete Math. 84 (1990), no. 1, 87-100.  ...  The author establishes necessary and sufficient conditions for a countably infinite graph G to be the union of infinitely many edge- disjoint two-way infinite Hamiltonian paths.  ... 

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

Recent trends and future directions in vertex-transitive graphs

Klavdija Kutnar, Dragan Marušič
2008 Ars Mathematica Contemporanea  
Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs  ...  , are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over  ...  Acknowledgment The authors wish to thank Aleksander Malnič andŠtefko Miklavič, and the referees for helpful suggestions about the material in this article.  ... 
doi:10.26493/1855-3974.81.679 fatcat:ptk23ciowngfxlf7et5frp77eq

Page 359 of Mathematical Reviews Vol. 55, Issue 2 [page]

1978 Mathematical Reviews  
An odd factor of a graph G is a subgraph in which every vertex has odd valency (and they exist if and only if every component of G has even order).  ...  and almost transi- tive ones in the infinite case.  ... 

Page 1594 of Mathematical Reviews Vol. 57, Issue 5 [page]

1979 Mathematical Reviews  
Hamilton graphs (pp. 545-556); Carsten Thomas- sen, Hypohamiltonian graphs and digraphs (pp. 557-574); William T.  ...  It is shown that for each positive integer n, the complete graph on 2n vertices is the only nm-decomposable graph of constant valence for which equality holds above.  ... 

Uncoverings on graphs and network reliability [article]

Robert F. Bailey, Brett Stevens
2011 arXiv   pre-print
We construct examples of these for some infinite families of graphs. Many of these infinite families utilise factorisations or decompositions of graphs.  ...  In every case the size of the uncovering-by-bases is no larger than the number of edges in the graph and we conjecture that this may be true in general.  ...  Acknowledgements The authors wish to thank NSERC and the Ontario Ministry of Research and Innovation for their financial support, and numerous colleagues for useful discussions.  ... 
arXiv:1012.2928v2 fatcat:iz7ditw5qvdz3phhymprhw67yu

Page 5466 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
Hirasaka and M. Muzychuk, Association schemes generated by a non-symmetric relation of valency 2 (109-135); Milan Hlad- nik, Dragan MaruSi¢ and Tomaz Pisanski, Cyclic Haar graphs (137-152); K. T.  ...  ¢éek, A note on degree conditions for Hamiltonicity in 2-connected claw-free graphs (253-268).  ... 

Connection Matrices and the Definability of Graph Parameters

Tomer Kotek, Johann Makowsky, Arnaud Durand
2014 Logical Methods in Computer Science  
We demonstrate its vast applicability in simplifying known and new non-definability results of graph properties and finding new non-definability results for graph parameters.  ...  In this paper we extend and prove in detail the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with counting (CMSOL) from B. Godlin, T.  ...  Graphs with average degree at most |V | 2 . Aperiodic digraphs (where the greatest common divisor of the lengths of all cycles in the graph is 1).  ... 
doi:10.2168/lmcs-10(4:1)2014 fatcat:us4rp5u2abaxtji7gyqra347ly

A SURVEY OF PROGRESS IN GRAPH THEORY IN THE SOVIET UNION

James Turner, William H. Kautz
1970 Annals of the New York Academy of Sciences  
Very littlv Soviet work has been reported on connectivity propertieb of graphs, matroid theory, the exact enumeration of graphs having prescribed properties, isomorphism testing, graph coloring, and the  ...  The best Sovic work has been concerned with bounds on numerical indices associated with graphs, properties of algebraic stvuctures associated with graphs, and operations on graphs.  ...  They give methods for finding maximal complete subgr'phs of a graph, Hamilton cycles of a digraph, and Hamilton circuits of a graph. The proceaures are very inefficient computationally.  ... 
doi:10.1111/j.1749-6632.1970.tb56496.x fatcat:cbb5x22w2farrnw5sijyovn66i

Master index of volumes 71–80

1990 Discrete Mathematics  
., Hamiltonian paths in Cayley digraphs of finitely-generated infinite Abelian groups Kabadi, S.N., see Chandrasekaran, R. Kajitani, Y., S. Ueno and H.  ...  Quaife, Minimum graphs of specified diameter, connectivity and valence. II ErdBs, P., Problems and results in combinatorial analysis and graph theory Erdiis, P. and A.  ... 
doi:10.1016/0012-365x(90)90304-z fatcat:tp4gu4hk2fbrdawvvu4vcsupii

Hamilton cycles in dense vertex-transitive graphs

Demetres Christofides, Jan Hladký, András Máthé
2014 Journal of combinatorial theory. Series B (Print)  
A famous conjecture of Lov\'asz states that every connected vertex-transitive graph contains a Hamilton path.  ...  In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.  ...  Acknowledgments The idea of using the LP-duality in conjunction with the Regularity Lemma originated in discussion of JH with Dan Král' and Diana Piguet on another (yet unpublished) project.  ... 
doi:10.1016/j.jctb.2014.05.001 fatcat:ki45v5onbzbxhpds6q3sr2oj2e

Cubic Cayley Graphs and Snarks [chapter]

Ademir Hujdurović, Klavdija Kutnar, Dragan Marušič
2014 Fields Institute Communications  
Combining the theory of Cayley maps with the existence of certain kinds of independent sets of vertices in arc-transitive graphs, some new partial results are obtained suggesting promising future research  ...  The well-known conjecture that there are no snarks amongst Cayley graphs is considered.  ...  The second author was supported in part by ARRS, P1-0285, J1-2055 and Z1-4006, and by ESF EuroGiga GReGAS.  ... 
doi:10.1007/978-1-4939-0781-6_2 fatcat:3zeawbc6lbdzpfomiwkmrxygwa
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