Bicolored graph partitioning, or: gerrymandering at its worst.

This study is motivated by an electoral application where we look into the following question: how much biased can the assignment of parliament seats be in a majority system under the effect of vicious gerrymandering when the two competing parties have the same electoral strength? To give a first theoretical answer to this question, we introduce a stylized combinatorial model, where the territory is represented by a rectangular grid graph, the vote outcome by a ''balanced'' red/blue node bicoloring and a district map by a connected partition of the grid whose components all have the same size. We constructively prove the existence in cycles and grid graphs of a balanced bicoloring and of two antagonist ''partisan'' district maps such that the discrepancy between their number of ''red'' (or ''blue'') districts for that bicoloring is extremely large, in fact as large as allowed by color balance.