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Longest cycles in r-regular r-connected graphs

Brad Jackson, T.D Parsons
1982 Journal of combinatorial theory. Series B (Print)  
It is shown that there is an integer N(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.  ... 
doi:10.1016/0095-8956(82)90001-6 fatcat:gyg64ezlsvgetijk2na2aie7mu

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]

Carol T. Zamfirescu
2022 arXiv   pre-print
It is shown that if a 3-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).  ... 
arXiv:2104.10020v2 fatcat:3lnped7sebch7avfwpmesvjbci

Longest cycles in $3$-connected $3$-regular graphs

J. A. Bondy, M. Simonovits
1980 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
On the other hand, if the length of a 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.  ... 
doi:10.4153/cjm-1980-076-2 fatcat:d3udp3pnlbck7dmwdjcduyybbe

A nonhamiltonian five-regular multitriangular polyhedral graph

Hansjoachim Walther
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.  ... 
doi:10.1016/0012-365x(95)00203-9 fatcat:ivqyhv3o4jevzk2yzoa4waizxe

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

Bill Jackson, Hao Li, Yongjin Zhu
1992 Discrete Mathematics  
Zhu, Dominating 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] .  ... 
doi:10.1016/0012-365x(92)90051-g fatcat:3clo6irqrzafflsidmk4rikwde

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

J.A Bondy, M Kouider
1988 Journal of combinatorial theory. Series B (Print)  
Let G be a 2-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.  ... 
doi:10.1016/0095-8956(88)90086-x fatcat:s6kvvrjtrngadovuzmprsjyc7i

Longest cycles in regular graphs

Genghua Fan
1985 Journal of combinatorial theory. Series B (Print)  
The paper is concerned with the 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.  ... 
doi:10.1016/0095-8956(85)90058-9 fatcat:m5fmtbirqrhjbfiftrm2viflwu

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.  ... 
doi:10.1016/0012-365x(89)90155-6 fatcat:3t3nb3wx6jdl5fr5kxim2ta7ma
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