The number of 2-edge-colored complete graphs with unique hamiltonian alternating cycle

Let G be a 2-edge-colored complete graph of even order n. A cycle of G is alternating if any two successive edges differ in color. We prove that there is a one-to-one correspondence between the set of bipartite tournaments of order n admitting a unique Hamiltonian cycle and the set of 2-edge-colored complete graphs of order n admitting a unique alternating Hamiltonian cycle.

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