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 thicknes θT(G) or caterpillar thicknes θC(G). If G has girth g with g ≥ 5, then θT(G) ≤ ⌊n/g⌋ + 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 θC, we prove that θC(G) ≤ ⌈(n - 2)/4⌉ when G has girth at least 6 and is not a 6-cycle. For triangle-free graphs, we conjecture that θC(G) ≤ ⌈3n/8⌉ and prove it for outerplanar graphs. For 2-connected graphs with girth g, we conjecture that θC(G) ≤ ⌊n/g⌋ when n ≥ max{6, g 2/2} and prove it for outerplanar graphs. All the bounds are sharp (sharpness in the ⌈3n/8⌉ bound is shown only for n ≡ 5 mod 8).