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For a graph $G$, the $t$-th power $G^t$ is the graph on $V(G)$ such that twovertices are adjacent if and only if they have distance at most $t$ in $G$; andthe $t$-th bi-power $G_B^t$ is the graph on $V(G)$ such that two vertices areadjacent if and only if their distance in $G$ is odd at most $t$. Fleischner'stheorem states that the square of every 2-connected finite graph has aHamiltonian cycle. Georgakopoulos prove that the square of every 2-connectedinfinite locally finite graph has a Hamiltonian circle. In this paper, weconsider the Hamiltonicity of the bi-power of bipartite graphs. We show thatfor every connected finite bipartite graph $G$ with a perfect matching, $G_B^3$has a Hamiltonian cycle. We also show that if $G$ is a connected infinitelocally finite bipartite graph with a perfect matching, then $G_B^3$ has aHamiltonian circle.

Hamiltonicity of bi-power of bipartite graphs, for finite and infinite cases