One sufficient condition for Hamiltonian graphs involving distances

Let $G$ be a 2-connected graph of order $n$ such that $2|N(x)\cup N(y)|+d(x)+d(y)\geq2n-1$ for each pair of nonadjacent vertices $x,y$. Then, as was proved in 1990 by G. T. Chen, $G$ is Hamiltonian. In this paper we introduce one more condition and prove that if $G$ is a 2-connected graph of order $n$ and $2|N(x)\cup N(y)|+d(x)+d(y)\geq2n-1$ for each pair of nonadjacent vertices $x,y$ such that $d(x,y)=2$, then $G$ is Hamiltonian.