Just to follow up @IlyaBogdanov's observation: a $k=4$ partition, whose
pieces pairwise share an edge, can be achieved (below), corresponding to a planar embedding of $K_4$.
But realizing the same for $k=5$ would
would lead to planar embedding of $K_5$, which is not possible.

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     [![K4][1]][1]


  [1]: https://i.sstatic.net/K0lEE.jpg