### Tree-Thickness and Caterpillar-Thickness under Girth Constraints

#### Abstract

We study extremal problems for decomposing a connected $n$-vertex graph $G$ into trees or into caterpillars. The least size of such a decomposition is the *tree thickness* $\theta_{\bf T}(G)$ or *caterpillar thickness* $\theta_{\bf C}(G)$. If $G$ has girth $g$ with $g\ge 5$, then $\theta_{\bf T}(G)\le \lfloor{n/g}\rfloor+1$. We conjecture that the bound holds also for $g=4$ and prove it when $G$ contains no subdivision of $K_{2,3}$ with girth 4. For $\theta_{\bf C}$, we prove that $\theta_{\bf C}(G)\le\lceil{(n-2)/4}\rceil$ when $G$ has girth at least $6$ and is not a $6$-cycle. For triangle-free graphs, we conjecture that $\theta_{\bf C}(G)\le\lceil{3n/8}\rceil$ and prove it for outerplanar graphs. For $2$-connected graphs with girth $g$, we conjecture that $\theta_{\bf C}(G)\le \lfloor{n/g}\rfloor$ when $n\ge\max\{6,g^2/2\}$ and prove it for outerplanar graphs. All the bounds are sharp (sharpness in the $\lceil{3n/8}\rceil$ bound is shown only for $n\equiv 5$ mod $8$).