# Extension of several sufficient conditions for Hamiltonian graphs

• Volume: 26, Issue: 1, page 23-39
• ISSN: 2083-5892

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## Abstract

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Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖u)|+d(u) ≥ n-1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇ 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph $\left({K}_{r}\cup Kₛ\cup Kₜ\right)\vee K₂$, 1 ≤ r ≤ s ≤ t or the graph $\left(\left(n+1\right)/2\right)K₁\vee {K}_{\left(n-1\right)/2}$. It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph satisfying the above condition. From this main result we derive a large number of extensions of previous sufficient conditions for hamiltonian graphs. All these results are sharp.

## How to cite

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Ahmed Ainouche. "Extension of several sufficient conditions for Hamiltonian graphs." Discussiones Mathematicae Graph Theory 26.1 (2006): 23-39. <http://eudml.org/doc/270300>.

@article{AhmedAinouche2006,
abstract = {Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖u)|+d(u) ≥ n-1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇ 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph $(K_r ∪ Kₛ ∪ Kₜ) ∨ K₂$, 1 ≤ r ≤ s ≤ t or the graph $((n+1)/2)K₁ ∨ K_\{(n-1)/2\}$. It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph satisfying the above condition. From this main result we derive a large number of extensions of previous sufficient conditions for hamiltonian graphs. All these results are sharp.},
author = {Ahmed Ainouche},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {hamiltonian graph; dual closure; neighborhood closure},
language = {eng},
number = {1},
pages = {23-39},
title = {Extension of several sufficient conditions for Hamiltonian graphs},
url = {http://eudml.org/doc/270300},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Ahmed Ainouche
TI - Extension of several sufficient conditions for Hamiltonian graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2006
VL - 26
IS - 1
SP - 23
EP - 39
AB - Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖u)|+d(u) ≥ n-1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇ 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph $(K_r ∪ Kₛ ∪ Kₜ) ∨ K₂$, 1 ≤ r ≤ s ≤ t or the graph $((n+1)/2)K₁ ∨ K_{(n-1)/2}$. It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph satisfying the above condition. From this main result we derive a large number of extensions of previous sufficient conditions for hamiltonian graphs. All these results are sharp.
LA - eng
KW - hamiltonian graph; dual closure; neighborhood closure
UR - http://eudml.org/doc/270300
ER -

## References

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