Abstract. The k-partition problem is as follows: Given a graph G and a positive integer k, partition the vertices of G into at most k parts A 1 , A 2 , . . . , A k , where it may be specified that A i induces a stable set, a clique, or an arbitrary subgraph, and pairs A i , A j (i = j) be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list k-partition problem generalizes the k-partition problem by specifying for each vertex x, a list L(x) of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list k-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete.
Let the lines of a complete graph be 3-colored so that no triangle gets 3 different ~olors. If two of these colors form perfect graphs then so does the third.
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