A graph $G$ is Hamiltonian-connected if there exists a Hamiltonian path between any two vertices of $G$. It is known that if $G$ is 2-connected then the graph $G^2$ is Hamiltonian-connected. In this paper we prove that the square of every self-complementary graph of order grater than 4 is Hamiltonian-connected. If $G$ is a $k$-critical graph, then we prove that the Mycielski graph $μ(G)$ is $(k+1)$-critical graph. Jarnicki et al.[7] proved that for every Hamiltonian graph of odd order, the Mycielski graph $μ(G)$ of $G$ is Hamiltonian-connected. They also pose a conjecture that if $G$ is Hamiltonian-connected and not $K_2$ then $μ(G)$ is Hamiltonian-connected. In this paper we also prove this conjecture.