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Consider the following reduction: $m$-colouring to $m$-partition.

Define an $m$-partition as: Given an undirected graph $G = (V, E)$ and an integer $j$. Does there exist a partition of the vertices into $m$ parts $\{V_1, V_2, \ldots , V_m\}$ such that at least $j$ of the edges have their endpoints in different parts of the partition?

I was able to come up with the reduction, but now I'm stuck at this question:

What I need to find is the values of $m$ where the reduction implies that $m$-Partition is NP-complete?

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  • $\begingroup$ For what values of m is m-coloring NP-complete? What research have you done on that? $\endgroup$ – D.W.♦ Apr 26 '20 at 7:17
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    $\begingroup$ @Matthew Not quite: the problem is still NP-complete for 4 colors. When you say "a map" you mean a planar graph and that's true, but in general it's still hard. $\endgroup$ – Juho Apr 26 '20 at 9:14
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    $\begingroup$ @Matthew No, consider the complete graph on 5 vertices, or more generally any graph with a large clique. $\endgroup$ – Juho Apr 27 '20 at 4:34
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    $\begingroup$ @Matthew To blow your mind even more, they are not the same problems. A graph without a large clique can still require a lot of colors :-) Have a look at the Mycielskians. $\endgroup$ – Juho Apr 27 '20 at 8:09
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    $\begingroup$ @Matthew Do as many exercises as you can, there's no substitute for that. Play around with graphs and reductions and you will improve over time. I don't have a particular resource in mind - there's a lot of books and other material that is good. $\endgroup$ – Juho Apr 28 '20 at 7:18
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The $1$-partition problem is trivial, since there is only one possible partition.

The $2$-partition problem is the same as MAX-CUT, which is known to be NP-complete.

For $m > 2$, the $m$-partition problem is NP-complete by reduction from $m$-coloring, which is NP-complete for that range of $m$.

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  • $\begingroup$ so m>=2 is your answer? $\endgroup$ – Objori Apr 26 '20 at 14:33
  • $\begingroup$ My answer is just what I wrote. $\endgroup$ – Yuval Filmus Apr 26 '20 at 14:35

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