Homology tests for graph colorings

TL;DR

This paper introduces a homological testing method for detecting obstructions to graph colorings, leveraging Hom-complexes and Stiefel-Whitney height to provide new algebraic tools for graph theory.

Contribution

It presents a novel homological framework combining Hom-complexes with Stiefel-Whitney height to create multiple tests for graph coloring obstructions.

Findings

Multiple homology tests vary with the choice of test graph

The tests can detect obstructions to graph colorings

Examples demonstrate the effectiveness of the tests

Abstract

We describe a simple homological test for obstructions to graph colorings. The main idea is to combine the framework of Hom-complexes with the following general fact: an arbitrary Z_2-space has nontrivial homology with Z_2-coefficients in the dimension equal to its Stiefel-Whitney height. Actually, as a result we have a whole family of homology tests, one for each test graph. In general, these tests will give different answers, depending heavily on the choice of the test graph. We illustrate this phenomenon with some examples.