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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.  ...

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

X. Liu, B. Wei
1997 Discrete Mathematics
In this paper we shall prove that ifmax{d(u): u ~ S} >~ n/2 holds for any 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

Kewen Zhao, Ronald J. Gould
2010 Colloquium Mathematicum
In 1994, Song 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).  ...

### 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.  ...

### 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

Roman Kužel, Kenta Ozeki, Kiyoshi Yoshimoto
2012 Discrete Mathematics
In this paper, we show that if G is an l-connected claw-free graph with minimum degree at least three 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.  ...

### A counterexample to the pseudo 2-factor isomorphic graph conjecture

Jan Goedgebeur
2015 Discrete Applied Mathematics
A graph G is 2-factor 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

A. Nagoor Gani, S. R. Latha
2016 SpringerPlus
A graph containing a 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]

Bart M.P. Jansen, László Kozma, Jesper Nederlof
2019 arXiv   pre-print
The results extend the range of tractability of the 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 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  ) 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]

Zh.G. Nikoghosyan
2014 arXiv   pre-print
In 1989, Bauer, Broersma Li 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.  ...

### Hamiltonian cycles in bipartite toroidal graphs with a partite set of degree four vertices

Jun Fujisawa, Atsuhiro Nakamoto, Kenta Ozeki
2013 Journal of combinatorial theory. Series B (Print)
We shall prove that G has a 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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