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Page 1510 of Mathematical Reviews Vol. , Issue 99c
[page]

1991
*
Mathematical Reviews
*

An

*independent**set*Z in G is called an*essential**independent**set*(or*essential**set*for simplicity) if there is {z),22} C Z such that dist(z),z2) = 2. ... G,, is a simple graph of order m,*and*J, is an*independent**set*of order m.” 99c:05128 05C45 Sierksma, Gerard (NL-GRON-SO; Groningen) Interchange graphs*and*the*Hamiltonian**cycle*polytope. ...##
###
Page 6462 of Mathematical Reviews Vol. , Issue 96k
[page]

1996
*
Mathematical Reviews
*

*independent*

*sets*

*and*

*Hamiltonian*

*cycles*. ... If we consider the

*essential*

*independent*

*sets*of order k + 1 instead of k in the assumption of the above statement, we can no longer assure the existence of a

*Hamiltonian*

*cycle*. ...

##
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A generalization of Bondy's and Fan's sufficient conditions for Hamiltonian graphs

1997
*
Discrete Mathematics
*

In this paper we shall prove that ifmax{d(u): u ~ S} >~ n/2 holds for any

doi:10.1016/0012-365x(95)00323-o
fatcat:ddi3mg4knrdehozh7lzj6n7u54
*essential**independent**set*S with k + 1 vertices of G, then either G is*hamiltonian*or G is one of three classes of exceptional ... Let S be an*independent**set*of G. S is called*essential*if there exists two distinct vertices in S which have a common neighbor in G. ... Ifmax{d(u): u ~ S} >>. n/2for any*essential**independent**set*S with k vertices in G, then G is*hamiltonian*. ...##
###
A note on the Song–Zhang theorem for Hamiltonian graphs

2010
*
Colloquium Mathematicum
*

In 1994, Song

doi:10.4064/cm120-1-5
fatcat:hqt5f5ygzbfjnmxib6cuar3hfy
*and*Zhang proved that if for each*independent**set*S of cardinality k + 1, one of the following condition holds: then G is*Hamiltonian*. ... We prove that if for each*essential**independent**set*S of cardinality k + 1, one of conditions (i) or (ii) holds, then G is*Hamiltonian*. ... Shao who provided many helpful suggestions,*and*Professor Hongjian Lai for his encouragement. The work of the first author was supported by the NSF of Hainan Province (no. 10501). ...##
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Page 778 of Mathematical Reviews Vol. , Issue 99b
[page]

1991
*
Mathematical Reviews
*

An

*independent**set*S of G is called*essential*if there exists {u,v} CS, such that dist(u,v) = 2. ... Zdenék Ryjaéek (Plzen) 99b:05096 05C45 05C35 05C55 Thomassen, Carsten (DK-TUD-M; Lyngby)*Independent*dominating*sets**and*a second*Hamiltonian**cycle*in regular graphs. (English summary) J. Combin. ...##
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Page 5183 of Mathematical Reviews Vol. , Issue 96i
[page]

1996
*
Mathematical Reviews
*

In this paper the
author proves that if UMS # @ for every

*essential**independent**set*S of order k +1 in G, then (1) G is*Hamiltonian*, or (2) G — u is*Hamiltonian*for some u € U, or (3) G belongs to one ... (PRC-ASBJ-AM; Beijing)*Independent**sets*, cliques*and**Hamiltonian*graphs. (English summary) Graphs Combin. 11 (1995), no. 3, 267-273. ...##
###
2-factors and independent sets on claw-free graphs

2012
*
Discrete Mathematics
*

In this paper, we show that if G is an l-connected claw-free graph with minimum degree at least three

doi:10.1016/j.disc.2011.08.020
fatcat:erof4jbzdfhhrloynfgbuclzva
*and*l ∈ {2, 3}, then for any maximum*independent**set*S, there exists a 2-factor in which each*cycle*... Acknowledgments The first author was supported by project 1M0545*and*Research Plan MSM 4977751301 of the Czech Ministry of Education. ... If δ ≥ n α ≥ 5, then there exist a maximum*independent**set*S*and*a 2-factor with α*cycles*such that each*cycle*contains exactly one vertex of S. 2. ...##
###
Page 49 of Mathematical Reviews Vol. , Issue 98A
[page]

1998
*
Mathematical Reviews
*

Moreover, each

*cycle*of G whose edge*set*is not contained in a fan contains at least two non-*essential*edges. ... (English*and*Chinese summaries) Adv. in Math. (China) 25 (1996), no. 1, 41-50. For a simple graph G, denote by .7,(G) the*set*of*independent**sets*of order ¢ of G. ...##
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A counterexample to the pseudo 2-factor isomorphic graph conjecture

2015
*
Discrete Applied Mathematics
*

A graph G is 2-factor

doi:10.1016/j.dam.2015.04.021
fatcat:trnijrwpuzanvas4yur74uqxqi
*hamiltonian*if all 2-factors of G are*hamiltonian**cycles*. ... Abreu et al. conjectured that K_3,3, the Heawood graph*and*the Pappus graph are the only*essentially*4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs (Abreu et al., Journal of Combinatorial ... Introduction*and*preliminaries All graphs considered in this paper are simple*and*undirected. Let G be such a graph. We denote the vertex*set*of G by V (G)*and*the edge*set*by E(G). ...##
###
A new algorithm to find fuzzy Hamilton cycle in a fuzzy network using adjacency matrix and minimum vertex degree

2016
*
SpringerPlus
*

A graph containing a

doi:10.1186/s40064-016-3473-x
pmid:27818892
pmcid:PMC5075334
fatcat:f6ds2hegizfojncq37sftc7yia
*Hamiltonian**cycle*is called a*Hamiltonian*graph. There have been several researches to find the number of*Hamiltonian**cycles*of a Hamilton graph. ... In this paper, a new algorithm is proposed to find fuzzy*Hamiltonian**cycle*using adjacency matrix*and*the degree of the vertices of a fuzzy graph. ... Acknowledgements The authors would like to thank the editor*and*anonymous referees of this journal for their constructive suggestions. ...##
###
Page 6768 of Mathematical Reviews Vol. , Issue 98K
[page]

1998
*
Mathematical Reviews
*

An

*essential**independent**set*of G is an*independent**set*such that the distance between each pair of vertices is 2. ... 98k:05096 98k:05096 05C38 05C45 A, Yongga |Ayongga] (PRC-IMTC; Hohhot) Degree sums*and**essential**independent**sets*in graphs. ...##
###
Hamiltonicity below Dirac's condition
[article]

2019
*
arXiv
*
pre-print

The results extend the range of tractability of the

arXiv:1902.01745v1
fatcat:uh57dh2zcfhatibtkjyl3xz4wq
*Hamiltonian**cycle*problem, showing that it is fixed-parameter tractable when parameterized below a natural bound. ... More precisely, we show that the*Hamiltonian**cycle*problem can be solved in time c^k · n^O(1), for some fixed constant c, if at least n-k vertices have degree at least n/2, or if all vertices have degree ... Acknowledgement We thank Naomi Nishimura, Ian Goulden,*and*Wendy Rush for obtaining a copy of Bondy's ...##
###
Some problems related to hamiltonian line graphs
[chapter]

2007
*
Proceedings of the International Conference on Complex Geometry and Related Fields
*

Part of this paper summarizes some of the recent developments in the study of

doi:10.1090/amsip/039/09
fatcat:pxde4ptk7jfptarou6gs2f5iqi
*hamiltonian*line graphs*and*the related*hamiltonian*claw-free graphs. ... The last section of this paper solves some problems on the*hamiltonian*like indices from a paper by Clark*and*Wormald in 1983. ... (Lai, Shao, Wu*and*Zhou [20] ) Every 3-connected,*essentially*11-connected line graph is*hamiltonian*. Conjecture 4 . 4 17. (H.-J. ...##
###
On Relative Length of Long Paths and Cycles in Graphs
[article]

2014
*
arXiv
*
pre-print

In 1989, Bauer, Broersma Li

arXiv:1407.8129v1
fatcat:zi63gn4ckfck7hgbztedqd2bhm
*and*Veldman proved that if G is a 2-connected graph with d(x)+d(y)+d(z)> n+κ for all triples x,y,z of*independent*vertices, then G is*hamiltonian*. ... Let G be a graph on n vertices, p the order of a longest path*and*κ the connectivity of G. ... A graph G is*hamiltonian*if G contains a Hamilton*cycle*, i.e. a*cycle*containing every vertex of G. A*cycle*C of a graph G is said to be dominating if V (G\C) is an*independent**set*. ...##
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Hamiltonian cycles in bipartite toroidal graphs with a partite set of degree four vertices

2013
*
Journal of combinatorial theory. Series B (Print)
*

We shall prove that G has a

doi:10.1016/j.jctb.2012.08.004
fatcat:644jj5vqaffrpgqlh4xxntmnji
*Hamiltonian**cycle*if (i) G is balanced, i.e., |X| = |Y |,*and*(ii) each vertex x ∈ X has degree four. ... This result implies that every 4-connected toroidal graph with toughness exactly one is*Hamiltonian*,*and*partially solves a well-known Nash-Williams' conjecture. ... Acknowledgments The authors are grateful to two anonymous referees for their careful reading of the paper*and*helpful suggestions for improving the presentation. ...
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