$\mathbb{Z}_{2}^{2}$-cordiality of complete and complete bipartite graphs

We prove that $K_{n}$ is $\mathbb{Z}_{2}^{2}$-cordial if and only if $1 \leq n \leq 3$ and that $K_{m,n}$ is $\mathbb{Z}_{2}^{2}$ if and only if it is false that $m=n=2$.