Dirac's minimum degree condition restricted to claws
A graph \(G\) of order \(n\) is 1-heavy (2-heavy) if at least one (two) of the end vertices of each induced claw \(K_{1,3}\) has degree at leat \(n/2\). It is shown that if \(G\) is a 2-connected 2-heavy graph of order \(n\geq 3\) such that \(|N(u)\cup N(v)|\geq 2\) for each pair of vertices \(u\) and \(v\) with \(d(u,v)=2\) and \(\max\{d(u),d(v)\}< n/2\), then \(G\) is Hamiltonian. This generalizes several known Hamiltonian results, including the result of Shi. Other forms of the result are also proved; for example, ``2-connected and 2-heavy'' can be replaced by ``3-connected and 1-heavy'' without changing the conclusion that the graph is Hamiltonian. Results on the relationship between the previously mentioned conditions and toughness and the existence of perfect matchings are also proved.
scientific article; zbMATH DE number 1018510
scientific article