Hamiltonicity of 2-connected quasi-claw-free graphs (Q1827784)

The author calls a graph quasi-claw-free if, for any two vertices \(x,y\) of distance \(2\), there exists a common neighbor \(u\) of \(x,y\) such that each neighbor of \(u\) (distinct from \(x,y\)) is also a neighbor of either \(x\) or \(y\) (or both). He then proves that a quasi-claw-free graph with \(n\) vertices and minimum degree at least \(n/4\) is Hamiltonian unless it belongs to a restricted (completely specified) class of graphs.