The Hamiltonicity Of Two Classes Of Graphs

Hamiltonian problem is one of the major problems in graph theory. The neces-sary and su±cient condition of a connected Hamiltonian graph has not been found.Many scholars have been focusing on the Hamiltonian problem in some classifiedgraph.The hamiltonicity of claw-free graphs had attracted the attention of many graphtheorists in a period of time. In 1998, Ainouche first put forward the concept ofquasi-claw-free graph, many results on claw-free graphs can be extended to quasi-claw-free graphs. In 2009, H.J.Boersm and E.Vumar introduced the concept ofP₃-dominated graphs, every quasi-claw-free graph is P₃-dominated graph. In thepresent paper, we prove that if G is a 3-connected and P₃-dominated graph withjGj 6 5±? 4, then G is Hamiltonian.In 2006, M. Li et al proved every connected N2-locally connected graph is quad-rangularly connected, and proved the theorem: every quadrangularly connectedclaw-free graph without vertices of degree 1, which does not contain an inducedsubgraph H isomorphic to two graphs such that N1(x;G) of every vertex x of de-gree 4 in H is disconnected is hamiltonian in《Quadrangularly connected claw-freegraphs》. We modified some°aws in this paper. The statements of every connectedN2-locally connected graph is quadrangularly connected is not°awless and give anew proof. In contrast to the incompleteness of the condition this proposition, theterm of"without vertices of degree 1"is redundant. In fact, by definition, a quad-rangularly connected graph has no vertex of degree 1. We also notice that there aresome flaws in the proof of theorem, though the proof is essentially correct. Actuallywe think that there are some errors in Claims 2-4. We restate this theorem in aslightly different form and present a correction for the part of the proof.