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In this paper, we shall prove that any two Hamiltonian triangulations on the sphere with n ! 5 vertices can be transformed into each other by at most 4n À 20 diagonal flips, preserving the existence of Hamilton cycles. Moreover, using this result, we shall prove that at most 6n À 30 diagonal flips are needed for any two triangulations on the sphere with n(More)
A triangulation of a surface is irreducible if no edge can be contracted to produce a triangula-tion of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g ≥ 0 with b ≥ 0 boundaries is O(g +(More)
Let G be a 3-connected bipartite graph with partite sets X ∪ Y which is em-beddable in the torus. We shall prove that G has a Hamiltonian cycle if (i) G is blanced , i.e., |X| = |Y |, and (ii) each vertex x ∈ 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(More)
A circuit graph (G, C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most n−7 3 vertices of degree 3. Our estimation for the number of(More)