Algebraic characterization of uniquely vertex colorable graphs

The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, it is known that k-colorability of a graph G is equivalent to the condition 1∈IG,k for a certain ideal IG,k⊆k[x1,…,xn]. In this paper, we extend this result by proving a general decomposition theorem for IG,k. This theorem allows us to give an algebraic characterization of uniquely k-colorable graphs. Our results also give algorithms for testing unique colorability. As an application, we verify a counterexample to a conjecture of Xu concerning uniquely 3-colorable graphs without triangles.