Abstract
Given a graph G = ( V,E ) and a finite set L ( v ) at each vertex v ϵ V , the List Coloring problem asks whether there exists a function f : V → ⋃ v ϵ V L ( V ) such that (i) f ( v )ϵ L ( v ) for each v ϵ V and (ii) f ( u ) ≠ f ( v ) whenever u , v ϵ V and uv ϵ E . One of our results states that this decision problem remains NP-complete even if all of the followingconditions are met: (1) each set L ( v ) has at most three elements, (2) each “color” x ϵ⋃ v ϵ V L ( v ) occurs in at most three sets L ( v ), (3) each vertex v ϵ V has degree at most three, and (4) G is a planar graph. On the other hand, strengthening any of the assumptions (1)–(3) yields a polynomially solvable problem. The connection between List Coloring and Boolean Satisfiability is discussed, too.
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