On some intriguing problems in hamiltonian graph theory—a survey

Abstract

We survey results and open problems in hamiltonian graph theory centered around three themes: regular graphs, t-tough graphs, and claw-free graphs.

How Many Conjectures Can You Stand? A Survey

“…Many such results involve degree conditions and other neighborhood conditions. Such results have been surveyed in several papers (see, e.g., [12,23,28]). We do not want to discuss such conditions in this survey, but here is a connectivity-only result.…”

“…Although the 2-tough conjecture restricted to claw-free graphs is equivalent to Conjecture 1, it is beyond the scope of this survey. We refer the reader to [12] for more details. Inspired by these techniques, independently of [39] it has been shown in [14] that Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by a conclusion similar to the one in Conjecture 16.…”

Cycles in graphs and related problems

“…For more detailed information we refer to an excellent survey by Broersma [76], where the author gives a description of a very useful proof technique based on a variation of Woodall's hopping lemma [281].…”

“…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.…”