Obviously it can never be less than that. But it does not look trivial that this number can always be reached. Probably simplest way to prove is by induction. It is true for M=2, N=2 and if you add one more pro the minimum number always increases by N/2. Likewise if you add one more game for each pro, the minimum number always increases by M/2.
If M*N is odd (roundup case), you need an additional pro.

_________________
If you say no, Elwood and I will come here for breakfast, lunch, and dinner every day of the week.

There's an easy construction: put the pros in a line and have each play N/2 games with each neighbor if N even, and (N-1)/2 and (N+1)/2 with the two neighbors if N is odd. Then you can pair up the first and last with the remaining games for each if N is even or M is even. If they're both odd, one player will have one more game remaining than the other, so have one more game played against this player.

My comment about this series: it's quite good. I really enjoyed the comments and specially the Lee ChangHo and Go Seigen ones gave us hours of comments and fun replaying in our weekly meetings.

We also learned overall what to look for when seeing the games of a particular player.

These guidelines about the style make useful the review and allow start to understand how does these players see the game.

The new Master Play book is out, the seven pros covered are:

Sakata Eio, Takagawa Kaku, Fujisawa Shuko, Rin Kaiho, Nie Weiping, Ma Xiaochun, and Cho Hunhyun.

It's $24 and 269 pages, which means about 19 pages of analysis per game so I reckon it's a bit more streamlined than prior books.

@xyzer:
Do you know if this will be the new format moving forward if there are more volumes in the future? The last two did two pros each so it seems like it's moving towards broader coverage. I think it would be really neat to see an entire volume devoted to one of the current leading pros I hear so much about like Gu Li or Lee Sedol.

In China I saw big, thick, hardcover books about Gu Li and Lee Sedol with tons of commented games. Of course I didn't buy them, I don't know Chinese. But it would be cool to have something like that available.