# What is the complexity of vertex cover on k-partite graphs?

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.

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.