Coloring Discrete Manifolds

TL;DR

This paper investigates the chromatic properties of discrete manifolds, establishing bounds and relationships between graph coloring, manifold dimension, and topological structures, with implications for understanding manifold colorings.

Contribution

It introduces bounds on chromatic numbers of discrete manifolds and explores the structure of Fisk's variety O(G) in relation to manifold dimension and topology.

Findings

Chromatic number of discrete d-manifolds is between d+1 and 2(d+1).

Existence of spheres with specific chromatic numbers based on dimension and join operations.

Fisk's variety O(G) relates to the manifold's topology and can be a union of lower-dimensional manifolds.

Abstract

Discrete d-manifolds are classes of finite simple graphs which can triangulate classical manifolds but which are defined entirely within graph theory. We show that the chromatic number X(G) of a discrete d-manifold G is sandwiched between d+1 and 2(d+1). From the general identity X(A+B)=X(A)+X(B) for the join A+B of two finite simple graphs, it follows that there are (2k)-spheres with chromatic number X=(3k+1) and (2k-1)-spheres with chromatic number X=3k. Examples of 2-manifolds with X(G)=5 have been known since the pioneering work of Fisk. Current data support the that the ceiling function of 3(d+1)/2 could be an upper bound for all d-manifolds G, generalizing a conjecture of Albertson-Stromquist, stating X(G) is bounded above by 5 for all 2-manifolds. For a d-manifold, Fisk has introduced the (d-2)-variety O(G). This graph O(G) has maximal simplices of dimension (d-2) and correspond to complete complete subgraphs K_{d-1} of G for which the dual circle has odd cardinality. In general, O(G) is a union of (d-2)-manifolds. We note that if O(S(x)) is either empty or a (d-3)-sphere for all x then O(G) is a (d-2)-manifold or empty. The knot O(G) is already interesting for 3-manifolds G because Fisk has demonstrated that every possible knot can appear as O(G) for some 3-manifold. For 4-manifolds G especially, the Fisk variety O(G) is a 2-manifold in G as long as all O(S(x)) are either empty or a knot in every unit 3-sphere S(x).