Graph colorings, flows and arithmetic Tutte polynomial

TL;DR

This paper introduces arithmetic colorings and flows on labeled graphs, linking their polynomials to the arithmetic Tutte polynomial, thereby generalizing classical graph theory results.

Contribution

It extends the concepts of colorings and flows to an arithmetic setting and connects their polynomials to the arithmetic Tutte polynomial, broadening classical graph theory.

Findings

Arithmetic chromatic and flow polynomials are specializations of the arithmetic Tutte polynomial.

Generalizes classical Tutte polynomial results to arithmetic colorings and flows.

Provides a unified framework for graph invariants with arithmetic structures.

Abstract

We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and arithmetic flow polynomial are given by suitable specializations of the associated arithmetic Tutte polynomial, generalizing classical results of Tutte.