An Algebraic Characterization of Uniquely Vertex Colorable Graphs
نویسندگان
چکیده
The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, k-colorability of a graph can be characterized in terms of whether its graph polynomial is contained in a certain ideal. In this paper, we interpret unique colorability in an analogous manner and prove an algebraic characterization for uniquely k-colorable graphs. Our result also gives algorithms for testing unique colorability. As an application, we verify a counterexample to a conjecture of Xu concerning uniquely 3-colorable graphs without triangles.
منابع مشابه
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 giv...
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