No, it's really that easy: Clearly the complete coverings with dominoes biject to matchings of the graph with vertices = squares and edges defined by 2-sets of

No, it's really that easy: Clearly the complete coverings with dominoes biject to matchings of the graph with vertices = squares and edges defined by 2-sets of squares sharing an edge on the chessboard. Clearly this graph is bipartite, but the two elements of the bipartision are of different cardinality. Thus no perfect matching can exist.