TITLE: Variant of Graph coloring
QUESTION [6 upvotes]: This is a problem came from social network analysis.
In a vertex colored (need not be proper) graph, an edge is monochromatic, if both endpoints of the edge are colored with the same color. Given a partially colored network, my goal is to extend it to total coloring such that the number of monochromatic edges in the network is maximum. 
If the network (graph) is complete graph then the problem is easy (just see which color is appearing more and assign it all uncolored vertices). Suppose if my graph is a threshold graph, then I don't have any idea how to solve it in polynomial time.
Can someone help me?

REPLY [6 votes]: Update 2017.09.26: As I've just learnt from Tamas Kiraly, this notion is well-studied and has several names, such as k-way cut, k-terminal cut, multiway cut. For at least three colors, the problem becomes $NP$-complete for general graphs, see E. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour, and M. Yannakakis: The complexity of multiterminal cuts.
Earlier stuff: For two colors this can be solved in polynomial time for any graph, as it can be converted to a MinCut problem. Just connect all red vertices into a single vertex $s$, and all blue vertices into another single vertex $t$, obtaining a multigraph (or if you prefer, a graph with edge weights). The Minimum cut in this new graph between $s$ and $t$ gives the partition that gives the coloring maximizing the number of the monochromatic edges.