A Hamiltonian Result on \(L_1\)-Graphs

A graph \(G\) is called an \(L_1\)-graph if, for each triple of vertices \(x, y,$ and \(z\) with \(d(x,y) = 2\) and \(z \in N(x) \cap N(y)\), \(d(x) + d(y) \geq |N(x) \cup N(y) \cup N(z)| – 1\). Let \(G\) be a \(3\)-connected \(L_1\)-graph of order \(n \geq 18\). If \(\delta(G) \geq n/3\), then every pair of vertices \(u\) and \(v\) in \(G\) with \(d(u,v) \geq 3\) is connected by a Hamiltonian path of \(G\).