Abstract
A family of subtrees of a graphG whose edge sets form a partition of the edge set ofG is called atree decomposition ofG. The minimum number of trees in a tree decomposition ofG is called thetree number ofG and is denoted byτ(G). It is known that ifG is connected thenτ(G) ≤ ⌈|G|/2⌉. In this paper we show that ifG is connected and has girthg ≥ 5 thenτ(G) ≤ ⌊|G|/g⌋ + 1. Surprisingly, the case wheng = 4 seems to be more difficult. We conjecture that in this caseτ(G) ≤ ⌊|G|/4⌋ + 1 and show a wide class of graphs that satisfy it. Also, some special graphs like complete bipartite graphs andn-dimensional cubes, for which we determine their tree numbers, satisfy it. In the general case we prove the weaker inequalityτ(G) ≤ ⌊(|G| − 1)/3⌋ + 1.
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Truszczyński, M. The tree number of a graph with a given girth. Period Math Hung 19, 273–286 (1988). https://doi.org/10.1007/BF01848836
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DOI: https://doi.org/10.1007/BF01848836