Abstract

Plane integral drawings of the platonic solid graphs with triangle faces

Highlights

  • Every planar graph can be drawn in the plane with noncrossing edges which are straight line segments (Steinitz and Rademacher, Wagner, Fary, Stein – see a short proof in [7])

  • It is an open problem [2, 4] whether the edges can be straight line segments of integral lengths

  • One can try to construct for each planar graph G such an integral plane drawing, denoted by D(G)

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Summary

Introduction

Every planar graph can be drawn in the plane with noncrossing edges which are straight line segments (Steinitz and Rademacher, Wagner, Fary, Stein – see a short proof in [7]) It is an open problem [2, 4] whether the edges can be straight line segments of integral lengths. For the five platonic solids, the tetrahedron, octahedron, cube, dodecahedron, and icosahedron, the minimum diameters of their integral plane graphs have been determined in [5] to be 17, 2, 13, 2, and 159, respectively Those three of them which are triangulations of the plane, that is, where the corresponding polyhedra have triangular faces, namely the tetrahedron, octahedron, and icosahedron, occur in connection with generalized matchstick graphs in [1].

Tetrahedron
Octahedron
Icosahedron
Concluding remarks
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