No Quantum Brooks' Theorem

TL;DR

This paper introduces quantum graph theory, constructs a family of quantum graphs with high chromatic numbers, and demonstrates fundamental differences between quantum and classical graph colorings.

Contribution

It develops bounds on quantum graph invariants, constructs a versatile family of quantum graphs, and highlights key differences from classical graph coloring.

Findings

Constructed quantum graphs with arbitrarily high chromatic numbers.

Derived bounds on quantum graph invariants using a step logarithm function.

Showed quantum graph colorings differ fundamentally from classical graph colorings.

Abstract

First, I introduce quantum graph theory. I also discuss a known lower bound on the independence numbers and derive from it an upper bound on the chromatic numbers of quantum graphs. Then, I construct a family of quantum graphs that can be described as tropical, cyclical, and commutative. I also define a step logarithm function and express with it the bounds on quantum graph invariants in closed form. Finally, I obtain an upper bound on the independence numbers and a lower bound on the chromatic numbers of the constructed quantum graphs that are similar in form to the existing bounds. I also show that the constructed family contains graphs of any valence with arbitrarily high chromatic numbers and conclude by it that quantum graph colorings are quite different from classical graph colorings.