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arXiv · 2503.23648

The spanning tree spectrum: improved bounds and simple proofs

Noga Alon·Matija Bucić·Lior Gishboliner
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Abstract

The number of spanning trees of a graph $G$, denoted $\tau(G)$, is a well studied graph parameter with numerous connections to other areas of mathematics. In a recent remarkable paper, answering a question of Sedl\'a\v{c}ek from 1969, Chan, Kontorovich and Pak showed that $\tau(G)$ takes at least $1.1103^n$ different values across simple (and planar) $n$-vertex graphs $G$, for large enough $n$. We give a very short, purely combinatorial proof that at least $1.55^n$ values are attained. We also prove that exponential growth can be achieved with regular graphs, determining the growth rate in another problem first raised by Sedl\'a\v{c}ek in the late 1960's. We further show that the following modular dual version of the result holds. For any integer $N$ and any $u < N$ there exists a planar graph on $O(\log N)$ vertices whose number of spanning trees is $u$ modulo $N$.

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Noga Alon, Matija Bucić, Lior Gishboliner. 2025-03-31. The spanning tree spectrum: improved bounds and simple proofs. https://arxiv.org/abs/2503.23648

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