  The famous Wegner's Planar Graph Conjecture asserts tight upper bounds on the
chromatic number of the square $G^2$ of a planar graph $G$, depending on the
maximum degree $\Delta(G)$ of $G$. The only case that the conjecture is
resolved is when $\Delta(G)=3$, which was proven to be true by Thomassen, and
independently by Hartke, Jahanbekam, and Thomas. For $\Delta(G)=4$, Wegner's
Planar Graph Conjecture states that the chromatic number of $G^2$ is at most 9;
even this case is still widely open, and very recently Bousquet, de Meyer,
Deschamps, and Pierron claimed an upper bound of 12.
  We take a completely different approach, and show that a relaxation of
properly coloring the square of a planar graph $G$ with $\Delta(G)=4$ can be
achieved with 9 colors. Instead of requiring every color in the neighborhood of
a vertex to be unique, which is equivalent to a proper coloring of $G^2$, we
seek a proper coloring of $G$ such that at most one color is allowed to be
repeated in the neighborhood of a vertex of degree 4, but nowhere else.
