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Page 7562 of Mathematical Reviews Vol. , Issue 98M [page]

1998 Mathematical Reviews  
The authors address the question, raised by Erdos, of how large the girth of a Hamiltonian weakly pancyclic graph can be.  ...  98m:05106 98m:05106 05C38 05C35 05C45 Bollobas, Béla (1-MEMP; Memphis, TN); Thomason, Andrew (4-CAMB; Cambridge) On the girth of Hamiltonian weakly pancyclic graphs. (English summary) J.  ... 

Weakly pancyclic graphs

Stephan Brandt, Ralph Faudree, Wayne Goddard
1998 Journal of Graph Theory  
In generalizing the concept of a pancyclic graph, we say that a graph is 'weakly pancyclic' if it contains cycles of every length between the length of a shortest and a longest cycle.  ...  In this paper it is shown that in many cases the requirements on a graph which ensure that it is weakly pancyclic are considerably weaker than those required to ensure that it is pancyclic.  ...  Part of the work was done while the first author was at the University of Twente and the third author was at the Massachusetts Institute of Technology  ... 
doi:10.1002/(sici)1097-0118(199803)27:3<141::aid-jgt3>3.0.co;2-o fatcat:yabfi6cmdnabbpmd23efk2ilcu

Weakly pancyclic graphs

Stephan Brandt, Ralph Faudree, Wayne Goddard
1998 Journal of Graph Theory  
In generalizing the concept of a pancyclic graph, we say that a graph is 'weakly pancyclic' if it contains cycles of every length between the length of a shortest and a longest cycle.  ...  In this paper it is shown that in many cases the requirements on a graph which ensure that it is weakly pancyclic are considerably weaker than those required to ensure that it is pancyclic.  ...  Part of the work was done while the first author was at the University of Twente and the third author was at the Massachusetts Institute of Technology  ... 
doi:10.1002/(sici)1097-0118(199803)27:3<141::aid-jgt3>3.3.co;2-d fatcat:7pzjkfxaqbclxj5hiaual2kyym

A note on cycles in locally Hamiltonian and locally Hamilton-connected graphs

Long Tang, Elkin Vumar
2018 Discussiones Mathematicae Graph Theory  
Let P be a property of a graph. A graph G is said to be locally P, if the subgraph induced by the open neighbourhood of every vertex in G has property P.  ...  Among other results, they proved that a connected, locally Hamiltonian graph G with maximum degree at least |V (G)| − 5 is weakly pancyclic.  ...  Acknowledgments The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which have greatly improved both the exposition and clarity of this paper.  ... 
doi:10.7151/dmgt.2124 fatcat:6slihnhnzzhk5m3wm3ckqsggxe

Weakly Pancyclic Graphs

Béla Bollobás, Andrew Thomason
1999 Journal of combinatorial theory. Series B (Print)  
A graph is called weakly pancyclic if it contains cycles of all lengths between its girth and circumference.  ...  From this, Brandt (1997) deduced that every non-bipartite graph of the stated order and size is weakly pancyclic.  ...  ACKNOWLEDGMENT We thank the referee for a careful reading and thoughtful comments.  ... 
doi:10.1006/jctb.1999.1916 fatcat:5cdc6szwejbgfph3rkyzpeijgy

Ramsey numbers of long cycles versus books or wheels

Lingsheng Shi
2010 European journal of combinatorics (Print)  
As consequences, Ramsey numbers of other sparse graphs are also obtained.  ...  number of a book B m = K 2 * K c m versus a cycle C n of order n satisfies R(B m , C n ) = 2n − 1 for n > (6m + 7)/4 which improves a result of Faudree et al., and the Ramsey number of a cycle C n versus  ...  Acknowledgements The author thanks Ralph Faudree for sending him an offprint [11] , Mihyun Kang for sending him the reference [3] , and a referee for simplifying the proof of Theorem 2.3.  ... 
doi:10.1016/j.ejc.2009.07.004 fatcat:fzdomgcvebazzgkeqenfufm3fy

Page 6602 of Mathematical Reviews Vol. , Issue 97K [page]

1997 Mathematical Reviews  
A graph is weakly pancyclic if it contains cycles of all lengths between its girth and circumference. Let 6 denote the minimum degree of a graph.  ...  In this paper it is shown that for a triangle-free, non-bipartite graph G of order n and independence number a, if 6 > n/3 then G is weakly pancyclic with c(G) = min{n, 2(n — a)} and girth 4, unless G  ... 

Cycles in k-traceable oriented graphs

Susan A. van Aardt, Jean E. Dunbar, Marietjie Frick, Morten H. Nielsen
2011 Discrete Mathematics  
However, we establish an upper bound (linear in k) on the order of k-traceable oriented graphs having a strong component with girth greater than 3.  ...  The present paper reports results pertaining to various cycle properties of strong k-traceable oriented graphs and explores the extent to which pancyclicity is retained by strong k-traceable oriented graphs  ...  All the authors wish to thank the University of South Africa and the NRF for financial support.  ... 
doi:10.1016/j.disc.2011.05.032 fatcat:5pd4ovorcfbdbek5rnkit5jfcy

Page 9584 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
2004m:05160 05 2004m:05160 05C45 Zhang, Jiaxiu; Li, Weixiao On Hamiltonian and pancyclic graph. (English summary) Southeast Asian Bull. Math. 26 (2003), no. 6, 1075-1083.  ...  on both sides of the cut. We illustrate that, for large enough val- ues of g, this threshold is of the correct order by constructing a graph with girth at least g, minimum degree 6, uw = Q(g~!  ... 

Advances on the Hamiltonian Problem - A Survey

Ronald J. Gould
2003 Graphs and Combinatorics  
For completeness of the presentation of both particular questions and the general area, it also contains material on closely related topics such as traceable, pancyclic and hamiltonian-connected graphs  ...  This article is intended as a survey, updating earlier surveys in the area.  ...  (b) Every nonbipartite graph with δ(G) ≥ (n + 2)/3 is weakly pancyclic (and has girth 3 or 4). Brandt [52] also considered other degree conditions for weakly pancyclic graphs.  ... 
doi:10.1007/s00373-002-0492-x fatcat:7isvoy4l6jfxxmeftkdcwoyhoy

Cycles in graphs and related problems

Antoni Marczyk
2008 Dissertationes Mathematicae  
Another question related to weakly pancyclic graphs was raised by Erdős. He asked how large the girth of a hamiltonian weakly pancyclic graph can be.  ...  In the same section we consider weakly pancyclic graphs, those that contain cycles of every length between their girth and circumference.  ...  By using the same method as in the proof of Lemma 6.32 we can show that F is pancyclic and contains a cycle of length p.  ... 
doi:10.4064/dm454-0-1 fatcat:nm4dk2d2xzdhvnhjt7t63uk6hu

The Ramsey numbers for cycles versus wheels of odd order

Yaojun Chen, T.C. Edwin Cheng, Zhengke Miao, C.T. Ng
2009 Applied Mathematics Letters  
Every non-bipartite graph G of order n with δ(G) ≥ (n + 2)/3 is weakly pancyclic with g(G) = 3 or 4. Lemma 3 (Dirac [7] ). Let G be a 2-connected graph of order n ≥ 3 with δ(G) = δ.  ...  For two given graphs G 1 and G 2 , the Ramsey number R(G 1 , G 2 ) is the smallest integer n such that for any graph G of order n, either G contains G 1 or the complement of G contains G 2 .  ...  A graph on n vertices is pancyclic if it contains cycles of every length l, 3 ≤ l ≤ n and weakly pancyclic if it contains cycles of every length l, g(G) ≤ l ≤ c(G).  ... 
doi:10.1016/j.aml.2009.07.014 fatcat:5s4q3lgrxff3zka324wsmjsmvi

Page 5762 of Mathematical Reviews Vol. , Issue 95j [page]

1995 Mathematical Reviews  
Veldman, Pancyclicity of Hamiltonian line graphs (379-385); Nguyen Van Ngoc [Nguyén Van Ngoc] and Zsolt Tuza, 4-chromatic graphs with large odd girth (387-392); Douglas B.  ...  and Yong Jin Zhu, Pancyclism in some K3-free graphs (281-286); You Xue Luo, Rong Si Chen and Fa Jie Li, Some results for cycle covers of graphs (287-293); Ji Xiang Meng and Qiong Xiang Huang, On the connectivity  ... 

Contents

2007 Discrete Mathematics  
Pelayo On the connectivity of cages with girth five, six and eight 1441 A. Nadolski The circular chromatic index of some class 2 graphs 1447 O. Pikhurko Trees are almost prime 1455  ...  PorschenOn generalizations of the shadow independent set problem 1473E. SidorowiczSize of weakly saturated graphs 1486 C. Balbuena and X.  ... 
doi:10.1016/s0012-365x(07)00083-0 fatcat:ppqyw5c6u5cmpnmx7zh4xihvgq

Global cycle properties in locally isometric graphs [article]

Adam Borchert, Skylar Nicol, Ortrud R. Oellermann
2016 arXiv   pre-print
with maximum degree at most 6 are weakly pancyclic.  ...  A graph G is locally isometric if the subgraph induced by the neighbourhood of every vertex is an isometric subgraph of G.  ...  A graph G that is not necessarily Hamiltonian but has cycles of every possible length from the shortest cycle length g(G) (called the girth of G) to the longest cycle length c(G) (called the circumference  ... 
arXiv:1506.03310v2 fatcat:n43ondhswjae3hfpteu5jmakda
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