Hamiltonian cycles in bipartite toroidal graphs with a partite set of degree four vertices

Let G be a 3-connected bipartite graph with partite sets [email protected]?Y which is embeddable in the torus. We shall prove that G has a Hamiltonian cycle if (i) G is balanced, i.e., |X|=|Y|, and (ii) each vertex [email protected]?X has degree four. In order to prove the result, we establish a result on orientations of quadrangular torus maps possibly with multiple edges. This result implies that every 4-connected toroidal graph with toughness exactly one is Hamiltonian, and partially solves a well-known [email protected]? conjecture.

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