Discussiones Mathematicae Graph Theory 27(2) (2007) 229-240
doi: 10.7151/dmgt.1357

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Ahmed Ainouche  and  Serge Lapiquonne

B.P. 7209, 97275 Schoelcher Cedex, Martinique FRANCE
e-mail: a.ainouche@martinique.univ-ag.fr
e-mail: s.lapiquonne@martinique.univ-ag.fr Abstract Given a 2-connected graph G on n vertices, let G* be its partially square graph, obtained by adding edges uv whenever the vertices u,v have a common neighbor x satisfying the condition NG(x) ⊆ NG[u]∪NG[v], where NG[x] = NG(x)∪{x}. In particular, this condition is satisfied if x does not center a claw (an induced K1,3). Clearly G ⊆ G* ⊆ G2, where G2 is the square of G. For any independent triple X = {x,y,z} we define


(X) = d(x)+d(y)+d(z)−| N(x)∩ N(y)∩N(z)| .

Flandrin et al. proved that a 2-connected graph G is hamiltonian if [`(σ)]3(X) ≥ n holds for any independent triple X in G. Replacing X in G by X in the larger graph G*, Wu et al. improved recently this result. In this paper we characterize the nonhamiltonian 2-connected graphs G satisfying the condition [`(σ)] 3(X) ≥ n−1 where X is independent in G*. Using the concept of dual closure we (i) give a short proof of the above results and (ii) we show that each graph G satisfying this condition is hamiltonian if and only if its dual closure does not belong to two well defined exceptional classes of graphs. This implies that it takes a polynomial time to check the nonhamiltonicity or the hamiltonicity of such G.

Keywords: cycles, partially square graph, degree sum, independent sets, neighborhood unions and intersections, dual closure.

2000 Mathematics Subject Classification: 05C38, 05C45.


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Received 23 September 2005
Revised 12 March 2007
Accepted 12 March 2007