A graph \(G\) is called an \(L_1\)-graph if, for each triple of vertices \(u\), \(v\), and \(w\) with \(d(u,v) = 2\) and \(w \in N(u) \cap N(v)\), the condition \(d(u) + d(v) > |N(u) \cup N(v) \cup N(w)| – 1\) holds. This paper presents two results on the hamiltonicity of \(L_1\)-graphs.