Minimal graphs with a prescribed number of spanning trees. (English) Zbl 1490.05127

Summary: For \(n\geq 3\), let \(\alpha(n)\) denote the minimum number of vertices in a graph with exactly \(n\) spanning subtrees. This notion was introduced by J. Sedláček [Can. Math. Bull. 13, 515–517 (1970; Zbl 0202.23501)] and has been studied by J. Azarija and R. Škrekovski [Math. Bohem. 138, No. 2, 121–131 (2013; Zbl 1289.05043)] In particular, they have conjectured that \(\alpha(n)=o(\log n)\). This paper will prove a weak version of this conjecture: specifically we will show that \(\alpha(n)=O((\log n)^{3/2}/(\log\log n))\). This bound is substantially larger than the conjectured upper bound; it is at least in the same ballpark.


05C35 Extremal problems in graph theory
05C05 Trees


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