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- W2949329486 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 $1 in I_{G,k}$ for a certain ideal $I_{G,k} subseteq k[x_1, ..., x_n]$. 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." @default.
- W2949329486 created "2019-06-27" @default.
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- W2949329486 date "2006-06-22" @default.
- W2949329486 modified "2023-09-27" @default.
- W2949329486 title "Algebraic Characterization of Uniquely Vertex Colorable Graphs" @default.
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