Fleischner's theorem says that the square of every 2-connected graph contains a Hamiltonian cycle. We present a proof resulting in an O(|E|) algorithm for producing a Hamiltonian cycle in the square G2 of a 2-connected graph G = (V;E). The previous best was O(|V |2) by Lau in 1980. More generally, we get an O(|E|) algorithm for producing a Hamiltonian path between any two prescribed vertices, and we get an O(|V |2) algorithm for producing cycles C3;C4; : : : ;C|V| in G2 of lengths 3; 4; : : : ; |V|, respectively. © Copyright 2018 by SIAM.