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 possibleimpossible.
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.
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 impossible.
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.
