PDF

Discussiones Mathematicae Graph Theory 21(1) (2001) 255-266
DOI: https://doi.org/10.7151/dmgt.1148

REMARKS ON PARTIALLY SQUARE GRAPHS, HAMILTONICITY AND CIRCUMFERENCE

Hamamache Kheddouci

LE2I FRE-CNRS 2309
Université de Bourgogne, B.P. 47870
21078 Dijon Cedex, France
e-mail: kheddouc@u-bourgogne.fr

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^º = 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^º and we improve some known results.

Keywords: partially square graph, claw-free graph, independent set, hamiltonicity and circumference.

2000 Mathematics Subject Classification: 05C45.

References

[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.
[2]	A. Ainouche and M. Kouider, Hamiltonism and partially square graph, Graphs and Combinatorics 15 (1999) 257-265, doi: 10.1007/s003730050059.
[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]	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]	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.

Received 28 December 2000
Revised 16 May 2001