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Algorithmic complexity of list colorings. (English) Zbl 0807.68055
Summary: Given a graph \(G = (V,E)\) and a finite set \(L(v)\) at each vertex \(v \in V\), the List Coloring problem asks whether there exists a function \(f : V \to \cup_{v \in V} L(v)\) such that (i) \(f(v) \in L(v)\) for each \(v \in V\) and (ii) \(f(u) \neq f(v)\) whenever \(u,v \in V\) and \(uv \in E\). One of our results states that this decision problem remains NP-complete even if all of the following conditions are met: (1) each set \(L(v)\) has at most three elements, (2) each “color” \(x \in \cup_{v \in V} L(v)\) occurs in at most three sets \(L(v)\), (3) each vertex \(v \in V\) has degree at most three, and (4) \(G\) is a planar graph. On the other hand, strengthening any of the three assumptions given in the paper yields a polynomially solvable problem. The connection between List Coloring and Boolean Satisfiability is discussed, too.

MSC:
68Q25 Analysis of algorithms and problem complexity
05C15 Coloring of graphs and hypergraphs
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
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