On Polyhedral Realization with Isosceles Triangles

TL;DR

This paper investigates the geometric properties of polyhedral graphs with triangular faces, proving the existence of convex polyhedra with scalene faces and identifying families with congruent isosceles faces, advancing understanding of face configurations.

Contribution

It constructs examples of convex polyhedra with scalene faces, introduces new families of congruent isosceles face polyhedra, and analyzes their graph properties.

Findings

Existence of convex polyhedra with at least one scalene face.

Identification of an infinite family of convex polyhedra with congruent isosceles faces.

Graphs of such polyhedra have bounded diameter and small dominating sets.

Abstract

Answering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene triangle. Our construction is based on Kleetopes, and shows that there exists an integer $i$ such that all convex $i$-iterated Kleetopes have a scalene face. However, we also show that all Kleetopes of triangulated polyhedral graphs have non-convex non-self-crossing realizations in which all faces are isosceles. We answer another question of Malkevitch by observing that a spherical tiling of Dawson (2005) leads to a fourth infinite family of convex polyhedra in which all faces are congruent isosceles triangles, adding one to the three families previously known to Malkevitch. We prove that the graphs of convex polyhedra with congruent isosceles faces have bounded diameter and have dominating sets of bounded size.