Tree-Thickness and Caterpillar-Thickness under Girth Constraints

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 θT(G) or caterpillar thickness θC(G). If G has girth g with g ≥ 5, then θT(G) ≤ bn/gc + 1. We conjecture that the bound holds also for g = 4 and prove it when G contains no subdivision of K2,3 with girth 4. For θC, we prove that θC(G) ≤ d(n − 2)/4e when G has girth at least 6 and is not a 6-cycle. For triangle-free graphs, we conjecture that θC(G) ≤ d3n/8e and prove it for outerplanar graphs. For 2-connected graphs with girth g, we conjecture that θC(G) ≤ bn/gc when n ≥ max{6, g2/2} and prove it for outerplanar graphs. All the bounds are sharp (sharpness in the d3n/8e bound is shown only for n ≡ 5 mod 8).