  A cyclic coloring of a plane graph $G$ is a coloring of its vertices such
that vertices incident with the same face have distinct colors. The minimum
number of colors in a cyclic coloring of a plane graph $G$ is its cyclic
chromatic number $\chi_c(G)$. Let $\Delta^*(G)$ be the maximum face degree of a
graph $G$.
  In this note we show that to prove the Cyclic Coloring Conjecture of Borodin
from 1984, saying that every connected plane graph $G$ has $\chi_c(G) \leq
\lfloor \frac{3}{2}\Delta^*(G)\rfloor$, it is enough to do it for subdivisions
of simple $3$-connected plane graphs.
  We have discovered four new different upper bounds on $\chi_c(G)$ for graphs
$G$ from this restricted family; three bounds of them are tight. As
corollaries, we have shown that the conjecture holds for subdivisions of plane
triangulations, simple $3$-connected plane quadrangulations, and simple
$3$-connected plane pentagulations with an even maximum face degree, for
regular subdivisions of simple $3$-connected plane graphs of maximum degree at
least 10, and for subdivisions of simple $3$-connected plane graphs having the
maximum face degree large enough in comparison with the number of vertices of
their longest paths consisting only of vertices of degree two.
