Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Relaxations of Ore’s condition on cycles. | Relaxations of Ore s condition on cycles Ahmed Ainouche CEREGMIA-GRIMAAG UAG-Campus de Schoelcher . 7209 97275 Schoelcher Cedex Martinique FRANCE Submitted Jun 14 2004 Accepted Jun 14 2006 Published Jul 28 2006 Abstract A simple undirected 2-connected graph G of order n belongs to class O n 0 if Ơ2 n . It is well known Ore s theorem that G is hamiltonian if 0 in which case the 2-connectedness hypothesis is implied. In this paper we provide a method for studying this class of graphs. As an application we give a full characterization of graphs G in O n 3 in terms of their dual hamiltonian closure. Keywords Hamiltonian Cycle Dual Closure. 1 Introduction We consider throughout only simple 2-connected graphs G V E . We let a G V G G denote respectively the independence number the matching number and the number of components of the graph G. A graph G is 1-tough if SI G S is true for any subset S c V with G S 1. For k a G we set ơk min ỵ S is a stable set . We use the term stable to mean independent set. A graph G of order n belongs to class O n 0 if Ơ2 n . It is well known 13 that G is hamiltonian if G 2 O n 0 in which case the 2-connectedness hypothesis is implied. Jung 8 proved that a 1-tough graph G 2 O n 11 4 is hamiltonian. Indeed this is a strong assumption which is not easy to verify since recognizing tough graphs is NP-Hard 10 . Ignoring the hypothesis of 1-toughness but conserving the constraint on n that is n 3 1 we obtained in 4 a characterization of graphs in O n 4 . Without any constraint on n a characterization of graphs in O n 2 is given in 2 and 9 . The same characterization was given by Schiermeyer 12 in terms of the dual-closure of G. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R60 1 In this paper we go a step further than Shiermeyer by giving a complete map of graphs in O n 3 with respect to the hamiltonian property. The dual closure 1 2 5 of those graphs is completely determined. This is indeed useful since then
