# Remarks on partially square graphs, hamiltonicity and circumference

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

top

## Abstract

top
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 ${N}_{G}\left(x\right)\subseteq {N}_{G}\left[u\right]\cup {N}_{G}\left[v\right]$, where ${N}_{G}\left[x\right]={N}_{G}\left(x\right)\cup x$. In the case where G is a claw-free graph, G* is equal to G². We define $\sigma °ₜ=min{\sum }_{x\in S}{d}_{G}\left(x\right):SisanindependentsetinG*and|S|=t$. We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.

## How to cite

top

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

top
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

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.