A generalization of a result of Häggkvist and Nicoghossian

Abstract

Using a variation of the Bondy-Chvbtal closure theorem the following result is proved: If G is a 2-connected graph with n vertices and connectivity K such that d(x) + d(y) + d(z) 2 n + K for any triple of independent vertices x, y, z, then G is hamiltonian.

Cycles in graphs and related problems

“…Another result involving the toughness and minimum degree was obtained by Bauer et al [22]. Theorem 3.77.…”

“…We have ∆/(2∆ − n) = 6. By Theorem 6.23, G has a cycle C p for every p belonging to the set Therefore, if an integer p belongs to this union [3,7] ∪ {9, 12, 13} ∪ [22,25], then the graph G has a cycle C p . Therefore, the solution of the problem is easily found.…”

“…It is worth noting that Bauer et al [29] showed that the problem of recognizing tough graphs is NP-hard. For surveys joining hamiltonian properties and toughness we recommend the articles by Bauer, Schmeichel and Veldman [34], Bauer, Broersma and Schmeichel [22,24] and Broersma [76].…”

“…We do not intend to give a complete survey of results related to the main topic. For the material not covered here, the reader is referred to the books by Walther and Voss [276], Voss [274], and the recent survey papers by Bondy [62], Gould [149,150], Broersma [76], Bauer, Broersma and Schmeichel [22] and Broersma, Ryjáček and Schiermeyer [80]. We recommend the books by Bollobás [49], [48], Alon and Spencer [8] and the article by Gould [149] to the reader interested in probabilistic methods in the research on cycles.…”

A degree sum condition for longest cycles in 3‐connected graphs

“…Let G be a 2-connected graph on n vertices. Then G contains a cycle of length at least min{σ Theorem 6 (Bauer et al [1]). Let G be a 2-connected graph on n vertices.…”