The fiber dimension of a graph

TL;DR

This paper introduces the concept of fiber dimension, a measure of how graphs can be embedded into integer lattice points within polytopes, and explores its properties and bounds.

Contribution

It defines fiber dimension for graphs, determines it for various classes, and provides an upper bound related to the chromatic number.

Findings

Any simple graph can be embedded as a fiber graph.

Fiber dimension is explicitly determined for certain graph classes.

An upper bound on fiber dimension in terms of chromatic number is established.

Abstract

Graphs on integer points of polytopes whose edges come from a set of allowed differences are studied. It is shown that any simple graph can be embedded in that way. The minimal dimension of such a representation is the fiber dimension of the given graph. The fiber dimension is determined for various classes of graphs and an upper bound in terms of the chromatic number is stated.