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Longest cycles in r-regular r-connected graphs

1982
*
Journal of combinatorial theory. Series B (Print)
*

It is shown that there is an integer N(

doi:10.1016/0095-8956(82)90001-6
fatcat:gyg64ezlsvgetijk2na2aie7mu
*r*, E) such that for all n ) N (if*r*is even) or for all even n ) N (if*r*is odd), there is an*r*-*connected**regular**graph*of valency*r*on exactly n vertices whose*longest*... That is, the number E > 0, no matter how small, is a "shortness coetlicient" for the family of*r*-valent*regular**r*-*connected**graphs*. ... Thomassen regarding our work on rcnh*graphs*. ...##
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Page 2007 of Mathematical Reviews Vol. , Issue 90D
[page]

1990
*
Mathematical Reviews
*

Some results on

*longest**cycles**in**regular*2-*connected**graphs*are also mentioned. {For the entire collection see MR 89g:05003.} George*R*. T. Hendry (4-ABER) ... A survey of results on*longest**cycles**in*2-*connected**graphs*with given minimum degree is presented. ...##
###
Page 1297 of Mathematical Reviews Vol. , Issue 91C
[page]

1991
*
Mathematical Reviews
*

(D-TUB)

*Longest**cycles*and independent sets*in*k-*connected**graphs*. Recent studies*in**graph*theory, 114-139, Vishwa, Gulbarga, 1989. ... This result is used to show that if every block of a*graph*G is Hamilton-*connected*, almost Hamilton-*connected*, or a*cycle*, then all*longest*paths*in*G have a vertex*in*common. ...##
###
Page 1389 of Mathematical Reviews Vol. , Issue 90C
[page]

1990
*
Mathematical Reviews
*

It is shown that the following three statements are equivalent: (i) Every cyclically 4-edge

*connected*4-*regular**graph*G has a*cycle*C if P.(G) #2, if P.(G) =, ... [Jackson, Bill] (4-LNDG) A note concerning some conjectures on cyclically 4-edge*connected*3-*regular**graphs*.*Graph*theory*in*memory of G. A. Dirac (Sandbjerg, 1985), 171-177, Ann. ...##
###
Regular graphs with few longest cycles
[article]

2022
*
arXiv
*
pre-print

It is shown that if a 3-

arXiv:2104.10020v2
fatcat:3lnped7sebch7avfwpmesvjbci
*regular**graph*G has a unique*longest**cycle*C, at least two components of G - E(C) have an odd number of vertices on C, and that there exist 3-*regular**graphs*with exactly two such ... We complement this by proving that the same conclusion holds for planar 4-*regular*3-*connected**graphs*, although it does not hold for planar 4-*regular*4-*connected**graphs*by a result of Brinkmann and Van ... The research presented*in*this paper was supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO). ...##
###
Longest cycles in $3$-connected $3$-regular graphs

1980
*
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
*

On the other hand, if the length of a

doi:10.4153/cjm-1980-076-2
fatcat:d3udp3pnlbck7dmwdjcduyybbe
*longest**cycle**in*P k is /, then P k+ i has no*cycle*of length greater than 8/ since a*cycle**in*P k+ i can visit at most I of the truncated Petersen*graphs*and can ... If G is a 3-*connected*^-*regular**graph*on n vertices, then it contains a*cycle*of length at least g( n ) = e c^£iï where â = § log e f. ...##
###
A nonhamiltonian five-regular multitriangular polyhedral graph

1996
*
Discrete Mathematics
*

*In*the class of all 5-

*regular*multitriangular polyhedral

*graphs*there exists a nonhamiltonian member with 1728 vertices, and moreover, this class has a shortness exponent smaller than one. ... Introduction Let v(G) and c(G) be the number of vertices of a

*graph*G(V, E) and the number of vertices

*in*a

*longest*

*cycle*of G, resp. ... Now, we combine three copies of A and three further copies of

*R*

*in*accordance with 391 The first two statements are easy to check. Let K be a

*longest*

*cycle*of T. ...

##
###
Page 5435 of Mathematical Reviews Vol. , Issue 87j
[page]

1987
*
Mathematical Reviews
*

A result proposed by the reviewer on Hamilton-

*connected**r*-*regular**graphs*G of order 2r, G # Kry [J.*Graph*Theory 7 (1983), no. 4, 429-436; MR 84k:05063], is extended to k-leaf-*connectedness*. ... The author shows that, if C is a*longest**cycle**in*a 2-*connected**graph*G with n vertices and minimum degree at least n/3, then G—C is either a complete*graph*or a set of independent vertices. ...##
###
Page 5750 of Mathematical Reviews Vol. , Issue 2003h
[page]

2003
*
Mathematical Reviews
*

length of a

*longest*path*in*a*graph*H. ... d.” 2003h:05118 05C38 05C40 Jung, Heinz Adolf (D-TUB; Berlin) Degree bounds for long paths and*cycles**in*k-*connected**graphs*. ...##
###
Dominating cycles in regular 3-connected graphs

1992
*
Discrete Mathematics
*

Zhu, Dominating

doi:10.1016/0012-365x(92)90051-g
fatcat:3clo6irqrzafflsidmk4rikwde
*cycles**in**regular*3-*connected**graphs*, Discrete Mathematics 102 (1992) 163-176. Let G be a 3-*connected*, k-*regular**graph*on at most 4k vertices. ... We show that, for k > 63, every*longest**cycle*of G is a dominating*cycle*. We conjecture that G is*in*fact hamiltonian. ... Let G be a 3-*connected*, k-*regular**graph*and C be a*longest**cycle**in*G. Then IV(C)1 2 min{]V(G)], 3k). We shall use the terminology of [5] . ...##
###
Page 636 of Mathematical Reviews Vol. , Issue 92b
[page]

1992
*
Mathematical Reviews
*

An (

*r*,m)-polygonal*graph*is an*r*-*regular**graph*G of girth m together with a set of m-*cycles*@ such that every path of length two is*in*a unique C € &. ... Summary: “We examine the problem of finding*longest**cycles**in*inner triangulations, that is, 2-*connected*planar*graphs**in*which all interior faces are triangles. ...##
###
Hamilton cycles in regular 2-connected graphs

1988
*
Journal of combinatorial theory. Series B (Print)
*

Let G be a 2-

doi:10.1016/0095-8956(88)90086-x
fatcat:s6kvvrjtrngadovuzmprsjyc7i
*connected**graph*with 6 3 k 3 3, let C be a*cycle**in*G, and let W be a component of G -C such that 2 6 1 V( W)l <k + 1. Then there is a*longest*path P*in*W which is strongly joined to C. ... PROOF OF THEOREM Let G be a 2-*connected*k-*regular**graph*with vertex set V, where 1 VI = n < 3k + 1. ... This research was funded*in*part by NSERC Operating Grant A7331. ...##
###
Longest cycles in regular graphs

1985
*
Journal of combinatorial theory. Series B (Print)
*

The paper is concerned with the

doi:10.1016/0095-8956(85)90058-9
fatcat:m5fmtbirqrhjbfiftrm2viflwu
*longest**cycles**in**regular*three-(or two-)*connected**graphs*. ...*In*particular, the following results are proved: (i) every 3-*connected*k-*regular**graph*on n vertices has a*cycle*of length at least min(3k, n); (ii) every 2-*connected*k-*regular**graph*on n vertices, where ... Choose a*longest**cycle*C so that the number of components*in**R*= G\C is minimal. ...##
###
Page 6434 of Mathematical Reviews Vol. , Issue 95k
[page]

1995
*
Mathematical Reviews
*

Hao Li (Orsay)
95k:05101 05C38 05C40
Li, Ming Chu (PRC-BST-MM; Beijing)

*Longest**cycles**in**regular*2-*connected*claw-free*graphs*. (English summary) Discrete Math. 137 (1995), no. 1-3, 277-295. ...*R*. (1- NORE; Boston, MA); Peled, U. N. (1-ILCC-MS; Chicago, IL)*Longest**cycles**in*threshold*graphs*. (English summary) Discrete Math. 135 (1994), no. 1-3, 169-176. ...##
###
T.D. Parsons — list of publications

1989
*
Discrete Mathematics
*

*Longest*

*cycles*

*in*

*r*-

*regular*

*r*-

*connected*

*graphs*(with B. Jackson), J. Combinat. Theory B32 (1982) 231-245. 28. On hamiltonian

*cycles*

*in*metacirculant

*graphs*(with B. ...

*Graph*Theory5 (1981) 55-77. 24. On

*r*-

*regular*

*r*-

*connected*non-hamiltonian

*graphs*(with B. Jackson), Bull. -Australian Math. Sot. 24 (1981) 205-220. 25. ...

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