The size of uniquely colorable graphs
Journal of Combinatorial Theory Series B
On uniquely 3-colorable graphs
Discrete Mathematics
Proceedings of the first Malta conference on Graphs and combinatorics
On small uniquely vertex-colourable graphs and Xu's conjecture
Discrete Mathematics
Kr-free uniquely vertex colorable graphs with minimum possible edges
Journal of Combinatorial Theory Series B
The division algorithm and the hilbert scheme
The division algorithm and the hilbert scheme
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Gröbner bases: a sampler of recent developments
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Hilbert's nullstellensatz and an algorithm for proving combinatorial infeasibility
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Algebraic properties of modulo q complete ℓ-wide families
Combinatorics, Probability and Computing
Expressing combinatorial problems by systems of polynomial equations and hilbert's nullstellensatz
Combinatorics, Probability and Computing
Some combinatorial applications of Gröbner bases
CAI'11 Proceedings of the 4th international conference on Algebraic informatics
Computing infeasibility certificates for combinatorial problems through Hilbert's Nullstellensatz
Journal of Symbolic Computation
Most Tensor Problems Are NP-Hard
Journal of the ACM (JACM)
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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.