Abstract
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 ∈ I G , k for a certain ideal I G , k ⊆ k [ x 1 , … , x n ] . In this paper, we extend this result by proving a general decomposition theorem for I G , 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.
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