Low Degree Nullstellensatz Certificates for 3-Colorability

TL;DR

This paper provides a simplified combinatorial characterization for detecting non-3-colorability of graphs using the NulLA algorithm over GF(2), and identifies the iteration at which this detection occurs for small graphs.

Contribution

It offers a direct combinatorial criterion for early detection of non-3-colorability in NulLA over GF(2), simplifying previous approaches.

Findings

Characterization of graphs detected in the first iteration of NulLA over GF(2)

Determination of detection iteration for all graphs up to 12 vertices

Simplification of the detection process without auxiliary directed graphs

Abstract

In a seminal paper, De Loera et. al introduce the algorithm NulLA (Nullstellensatz Linear Algebra) and use it to measure the difficulty of determining if a graph is not 3-colorable. The crux of this relies on a correspondence between 3-colorings of a graph and solutions to a certain system of polynomial equations over a field $\mathbb{k}$. In this article, we give a new direct combinatorial characterization of graphs that can be determined to be non-3-colorable in the first iteration of this algorithm when $\mathbb{k}=GF(2)$. This greatly simplifies the work of De Loera et. al, as we express the combinatorial characterization directly in terms of the graphs themselves without introducing superfluous directed graphs. Furthermore, for all graphs on at most $12$ vertices, we determine at which iteration NulLA detects a graph is not 3-colorable when $\mathbb{k}=GF(2)$.