Polyhedra without diagonals II
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POLYHEDRA WITHOUT DIAGONALS II ´ndor Szabo ´ Sa [Communicated by M´ aria B. Szendrei] Institute of Mathematics and Informatics, University of P´ecs Ifj´ us´ ag u. 6, H-7624 P´ecs, Hungary E-mail: [email protected] (Received June 19, 2008; Accepted January 6, 2009)
Abstract It is shown that no polyhedron without diagonals exists with five vertices.
1. Introduction This paper is a continuation of an earlier paper [3] with the same title. It is written to correct an argument in [3] by filling a gap pointed out by B. Gr¨ unbaum and G. C. Shephard [2]. The tetrahedron, as a solid body, is the convex hull of four points in the space that do not lie in one plane. It has four vertices, six edges, and four faces. Plainly the straight line sections determined by any two of the vertices are all edges of the tetrahedron and consequently the tetrahedron does not have any diagonal. The four faces of the tetrahedron form a surface that is homeomorphic to a sphere. It is a known fact that if the surface of a polyhedron is homeomorphic to a sphere and the polyhedron does not have any diagonal, then the polyhedron must be a tetrahedron. There is a polyhedron, the famous Cs´asz´ar polyhedron, whose surface is homoeomorphic to the torus and does not have any diagonal. It is an open problem if there is any further polyhedron without diagonals whose surface is homeomorphic to a sphere with a number of handles. On the other hand, among the polyhedra whose surfaces are not necessarily homeomorphic to a sphere with a number of handles there are many without diagonals. Namely, [3] exhibited a polyhedron without diagonals with c vertices for each c ≥ 6. Of course, c must be at least four and the tetrahedron provides an example for the case c = 4. There is a claim in [3] that there is no polyhedron without diagonals for c = 5. Mathematics subject classification numbers: 51H10, 51H20. Key words and phrases: polyhedron, Cs´ asz´ ar polyhedron. 0031-5303/2009/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
´ S. SZABO
182
To be more specific in [3] a concept of polyhedron, that slightly differs from the one we are familiar with from geometry similar to the one used mostly in algebraic topology, is introduced. Namely, a polyhedron is defined to be a union of finitely many tetrahedra that themselves form a simplicial decomposition of a connected bounded region in the space. As a consequence the concepts of interior, face, edge, vertex had to be redefined. The main result in [3] is the following. Theorem 1. Let c be an integer. (1) Each polyhedron with c = 5 vertices admits at least one diagonal. (2) For each c ≥ 6 there exists a polyhedron without diagonals having c vertices. In [3] statement (2) is proved by means of explicit constructions. It is clear from the constructions but for the sake of clarity we would like to reiterate that Theorem 1 holds only for polyhedra defined in [3]. In [3] the proof of statement (1) was based on the existence of a specific simplicial decomposition. However, in 1994, B. Gr¨ u
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