Graphs with large girth and chromatic number are hard for Nullstellensatz

TL;DR

This paper demonstrates that detecting non-colorability in graphs with large girth and chromatic number using Nullstellensatz-based linear algebra methods requires solving very large systems, indicating computational hardness.

Contribution

It establishes a lower bound on the size of linear systems needed for Nullstellensatz-based algorithms to certify non-colorability in certain complex graphs.

Findings

Large girth and chromatic number graphs require exponential-sized systems for detection

Nullstellensatz-based methods are computationally hard for these graphs

Lower bounds imply limitations of algebraic approaches in graph coloring

Abstract

We study the computational efficiency of approaches, based on Hilbert's Nullstellensatz, which use systems of linear equations for detecting non-colorability of graphs having large girth and chromatic number. We show that for every non-$k$-colorable graph with $n$ vertices and girth $g>4k$, the algorithm is required to solve systems of size at least $n^{\Omega(g)}$ in order to detect its non-$k$-colorability.