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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)

In this paper, we shall show that any two quadrangulations on any closed surface can be transformed into each other by diagonal slides and diagonal rotations if they have the same and sufficiently large number of vertices and if the homological properties of both quadrangulations coincide. 1996 Academic Press, Inc.

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)