2006/06/22 by Christopher J. Hillar, Hillar, Christopher J., Troels Windfeldt +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications #math.AC #math.CO
paper · pdf · doi:10.48550/arxiv.math/0606565
15 pages, 2 figures, print version, to appear J. Comb. Th. Ser. B
openalex publication_date 2006/06/22 · arxiv created 2007/09/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.