  An edge coloring of a graph $G$ is \emph{woody} if no cycle is monochromatic.
The \emph{arboricity} of a graph $G$, denoted by $\arb (G)$, is the least
number of colors needed for a woody coloring of $G$. A coloring of $G$ is
\emph{strongly woody} if after contraction of any single edge it is still
woody. In other words, not only any cycle in $G$ can be monochromatic but also
any \emph{broken cycle}, i.e., a simple path arising by deleting a single edge
from the cycle. The least number of colors in a strongly woody coloring of $G$
is denoted by $\zeta(G)$ and called the \emph{strong arboricity} of $G$.
  We prove that $\zeta(G)\leqslant \chi_a(G)$, where $\chi_a(G)$ is the
\emph{acyclic chromatic number} of $G$ (the least number of colors in a proper
vertex coloring without a $2$-colored cycle). In particular, we get that
$\zeta(G)\leqslant 5$ for planar graphs and $\zeta(G)\leqslant 4$ for
otuterplanar graphs. We conjecture that $\zeta(G)\leqslant 4$ holds for all
planar graphs. We also prove that $\zeta(G)\leqslant 4(\arb(G))^2$ holds for
arbitrary graph $G$. A natural generalziation of strong arboricity to
\emph{matroids} is also discussed, with a special focus on cographic matroids.
