[CrazyStat]

I haven’t seen this representation before—I suppose the vertices of the graph are the chessboard squares, the edges are adjacency (white squares can only be adjacent to black squares and vice versa, which gives the bipartite-ness), and covering two squares corresponds to removing those two vertices from the graph?

[praptak]

Yes, and this is a generalisation of the trick from the problem described in the article.

The chessboard in the article is a bipartite graph with different number of vertices in the two groups, so it cannot have a perfect matching.