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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

1998
*
Journal of Graph Theory
*

In generalizing

doi:10.1002/(sici)1097-0118(199803)27:3<141::aid-jgt3>3.0.co;2-o
fatcat:yabfi6cmdnabbpmd23efk2ilcu
*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 ...##
###
Weakly pancyclic graphs

1998
*
Journal of Graph Theory
*

In generalizing

doi:10.1002/(sici)1097-0118(199803)27:3<141::aid-jgt3>3.3.co;2-d
fatcat:7pzjkfxaqbclxj5hiaual2kyym
*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 ...##
###
A note on cycles in locally Hamiltonian and locally Hamilton-connected graphs

2018
*
Discussiones Mathematicae Graph Theory
*

Let P be a property

doi:10.7151/dmgt.2124
fatcat:6slihnhnzzhk5m3wm3ckqsggxe
*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. ...##
###
Weakly Pancyclic Graphs

1999
*
Journal of combinatorial theory. Series B (Print)
*

A

doi:10.1006/jctb.1999.1916
fatcat:5cdc6szwejbgfph3rkyzpeijgy
*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. ...##
###
Ramsey numbers of long cycles versus books or wheels

2010
*
European journal of combinatorics (Print)
*

As consequences, Ramsey numbers

doi:10.1016/j.ejc.2009.07.004
fatcat:fzdomgcvebazzgkeqenfufm3fy
*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. ...##
###
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

2011
*
Discrete Mathematics
*

However, we establish an upper bound (linear in k)

doi:10.1016/j.disc.2011.05.032
fatcat:5pd4ovorcfbdbek5rnkit5jfcy
*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. ...##
###
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

2003
*
Graphs and Combinatorics
*

For completeness

doi:10.1007/s00373-002-0492-x
fatcat:7isvoy4l6jfxxmeftkdcwoyhoy
*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*. ...##
###
Cycles in graphs and related problems

2008
*
Dissertationes Mathematicae
*

Another question related to

doi:10.4064/dm454-0-1
fatcat:nm4dk2d2xzdhvnhjt7t63uk6hu
*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. ...##
###
The Ramsey numbers for cycles versus wheels of odd order

2009
*
Applied Mathematics Letters
*

Every non-bipartite

doi:10.1016/j.aml.2009.07.014
fatcat:5s4q3lgrxff3zka324wsmjsmvi
*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). ...##
###
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

doi:10.1016/s0012-365x(07)00083-0
fatcat:ppqyw5c6u5cmpnmx7zh4xihvgq
*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. ...##
###
Global cycle properties in locally isometric graphs
[article]

2016
*
arXiv
*
pre-print

with maximum degree at most 6 are

arXiv:1506.03310v2
fatcat:n43ondhswjae3hfpteu5jmakda
*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 ...
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