Remarks on partially square graphs, hamiltonicity and circumference

Hamamache Kheddouci

Discussiones Mathematicae Graph Theory (2001)

  • Volume: 21, Issue: 2, page 255-266
  • ISSN: 2083-5892

Abstract

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Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition N G ( x ) ⊆ N G [ u ] ∪ N G [ v ] , where N G [ x ] = N G ( x ) ∪ x . In the case where G is a claw-free graph, G* is equal to G². We define σ ° ₜ = m i n ∑ x ∈ S d G ( x ) : S i s a n i n d e p e n d e n t s e t i n G * a n d | S | = t . We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.

How to cite

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Hamamache Kheddouci. "Remarks on partially square graphs, hamiltonicity and circumference." Discussiones Mathematicae Graph Theory 21.2 (2001): 255-266. <http://eudml.org/doc/270370>.

@article{HamamacheKheddouci2001,
abstract = {Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition $N_G(x) ⊆ N_G[u] ∪ N_G[v]$, where $N_G[x] = N_G(x) ∪ \{x\}$. In the case where G is a claw-free graph, G* is equal to G². We define $σ°ₜ = min\{ ∑_\{x∈S\} d_G(x):S is an independent set in G* and |S| = t\}$. We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.},
author = {Hamamache Kheddouci},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {partially square graph; claw-free graph; independent set; hamiltonicity and circumference; Hamiltonicity; circumference},
language = {eng},
number = {2},
pages = {255-266},
title = {Remarks on partially square graphs, hamiltonicity and circumference},
url = {http://eudml.org/doc/270370},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Hamamache Kheddouci
TI - Remarks on partially square graphs, hamiltonicity and circumference
JO - Discussiones Mathematicae Graph Theory
PY - 2001
VL - 21
IS - 2
SP - 255
EP - 266
AB - Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition $N_G(x) ⊆ N_G[u] ∪ N_G[v]$, where $N_G[x] = N_G(x) ∪ {x}$. In the case where G is a claw-free graph, G* is equal to G². We define $σ°ₜ = min{ ∑_{x∈S} d_G(x):S is an independent set in G* and |S| = t}$. We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.
LA - eng
KW - partially square graph; claw-free graph; independent set; hamiltonicity and circumference; Hamiltonicity; circumference
UR - http://eudml.org/doc/270370
ER -

References

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  1. [1] A. Ainouche, An improvement of Fraisse's sufficient condition for hamiltonian graphs, J. Graph Theory 16 (1992) 529-543, doi: 10.1002/jgt.3190160602. Zbl0770.05070
  2. [2] A. Ainouche and M. Kouider, Hamiltonism and partially square graph, Graphs and Combinatorics 15 (1999) 257-265, doi: 10.1007/s003730050059. Zbl0933.05096
  3. [3] J.C. Bermond, On Hamiltonian Walks, in: C.St.J.A. Nash-Wiliams and J. Sheehan, eds, Proceedings of the Fifth British Combinatorial Conference, Aberdeen, 1975 (Congr. Numerantium XV, Utilitas Math. Publ. Inc., 1975) 41-51. 
  4. [4] A. Bondy, Longest paths and cycles in graphs of high degree, Research report CORR 80-16 Dept of Combinatorics and Optimization (University of Waterloo, 1980). 
  5. [5] I. Fournier and P. Fraisse, On a conjecture of Bondy, J. Combin. Theory (B) 39 (1985) 17-26, doi: 10.1016/0095-8956(85)90035-8. Zbl0576.05035

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