On the Coherent Labelling Conjecture of a Polyhedron in Three Dimensions

TL;DR

This paper investigates the coherent labelling of three-dimensional polyhedra, classifies labellings for tetrahedra, and proves the conjecture for certain classes of polyhedra, revealing which are coherently labellable and which are not.

Contribution

It classifies all coherent labellings of tetrahedra, extends results to various polyhedra, and proves the conjecture for a significant class of constructible polyhedra.

Findings

All 48 coherent labellings of a tetrahedron are exhibited.

Certain simplicial polyhedra like bipyramids and gyroelongated bipyramids are coherently labellable.

The main conjecture is proven for polyhedra constructible from tetrahedra through specific attachments.

Abstract

In this article we consider an open conjecture about coherently labelling a polyhedron in three dimensions. We exhibit all the forty eight possible coherent labellings of a tetrahedron. We also exhibit that some simplicial polyhedra like bipyramids, Kleetopes, gyroelongated bipyramids are coherently labellable. Also we prove that pyramids over $n$-gons for $n\geq 4$, which are not simplicial polyhedra, are coherently labellable. We prove that among platonic solids, the cube and the dodecahedron are not coherently labellable, even though, the tetrahedron, the octahedron and the icosahedron are coherently labellable. Unlike the case of a tetrahedron, in general for a polyhedron, we show that a coherent labelling need not induce a coherent labelling at a vertex. We prove the main conjecture in the affirmative for a certain class of polyhedra which are constructible from tetrahedra through certain types of edge and face vanishing tetrahedron attachments. As a consequence we conclude that a cube cannot be obtained from only these type of tetrahedron attachments. We also give an obstruction criterion for a polyhedron to be not coherently labellable and consequentially show that any polyhedron obtained from a pyramid with its apex chopped off is not coherently labellable. Finally with the suggestion of the affirmative results we prove the main theorem that any simplicial polyhedron is coherently labellable.