Strong arboricity of graphs

Abstract

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 $\zeta (G)⩽{\chi }_a(G)$, 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 $\zeta (G)⩽5$ for planar graphs and $\zeta (G)⩽4$ for otuterplanar graphs. We conjecture that $\zeta (G)⩽4$ holds for all planar graphs. We also prove that $ζ(G)\leqslant 4(\arb(G))^2$ holds for arbitrary graph $G$. A natural generalziation of strong arboricity to \emph{matroids} is also discussed, with a special focus on cographic matroids.