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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.)
##### Keywords:
list coloring problem; Boolean satisfiability
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##### References:
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