$1$-skeleton of $S^2$ can't be graphs $K_5$ or bipartite graph $K_{3,3}$

For $K_5$ (I leave $K_{3,3}$ out, as it's a homework): it has $10$ edges, so your polygons have together $20$ edges (every edge of $K_5$ is supposed to come from gluing of two polygons along an edge). Euler characteristic gives you that the number of polygons is $7$. If $n_1,\dots,n_7$ are the numbers of edges of the polygons, since $n_1+\dots+n_7=20$, we can't have $n_i\geq 3$ for every $i$. An $i$ where $n_i=2$ (it's not really a polygon, but whatever) would mean that two of the vertices of $K_5$ are connected by two edges - and that's not the case.

($K_{3,3}$ is very similar - but there is a little change in the reasoning)