A proper vertex coloring of a graph $G$ is acyclic if there is no bicolored cycle. A graph is 2-degenerate if its every subgraph has a vertex of degree at most 2. I think every 2-degenerate graph has an acyclic proper vertex coloring using 3 colors, but I did not find any source stating this. Does anyone know this? Am I right? Thanks in advance.



Acyclic proper coloring of 2-degenerate graphs
Corollary 3 in <https://dmtcs.episciences.org/344> says that for every graph $G$, if $G'$ is the 1-subdivision of $G$, then the acyclic chromatic number of $G'$ is at least $\sqrt{\frac12 \chi(G)}$. Apply this result with $G$ the complete graph $K\_n$. Then the acyclic chromatic number of $K'\_n$ is at least $\sqrt{\frac12 n}$. Since $K'\_n$ is 2-degenerate, the acyclic chromatic number of 2-degenerate graphs is unbounded.


