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Given a k-partite graph which is already partitioned into k parts, what is the complexity of finding a vertex cover of minimum size?

I guess that it's NP-hard, but couldn't yet prove it or find reference for it. I'm also interested in the dependence on k.

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For bipartite graphs, vertex cover is polynomially solvable by routine techniques from matching theory.

For $k$-partite graphs with $k\ge3$, we observe the following:

  • Vertex cover is NP-complete on cubic graphs
  • By Brooks' theorem, every cubic graph (except $K_4$) is 3-colorable and hence 3-partite.
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