A sufficient condition for graphs with large neighborhood unions to be traceable Original Research Article

We prove that a 2-connected graph G of order p is traceable if |N(u)∪N(v)| − |N(w)∪N(x)| ⩾ p −1 for all 4-tuples {u, v, w, x} with d(u, v) = d(w, x) = 2 (u, v, w, x are distinct vertices of G). In addition, we give a short proof of Lindquesterʹs conjecture.