The Matching Number and Hamiltonicity of Quasi-Claw-Free Graphs

A graph \( G \) is quasi-claw-free if it satisfies the property: \( d(x, y) = 2 \) implies there exists \( u \in N(x) \cap N(y) \) such that \( N[u] \subseteq N[x] \cup N[y] \). The matching number of a graph \( G \) is the size of a maximum matching in the graph. In this note, we present a sufficient condition involving the matching number for the Hamiltonicity of quasi-claw-free graphs.