Publisher Summary This chapter focuses on Hamiltonian paths in squares of infinite locally finite blocks. The Fleischner's theorem is discussed, which states that the square of every finite block is Hamiltonian. This has subsequently been extended in various directions. For example, for any vertex x of a block G, G2 contains a Hamiltonian cycle C such that the two edges of C incident with x are edges of G. This Fleischner's theorem is presented for countable graphs. An obvious necessary condition for a graph to possess a one-way infinite Hamiltonian path is that the deletion of any finite vertex set results in graph with only one infinite component. This chapter answers affirmatively the question that arises when restricted to locally finite graphs.