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arXiv · 2608.11990

On spanning trees whose degrees are congruent to one modulo $\ell$

Zhidan Yan·Wei Wang
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Abstract

An $\ell$-congruent spanning tree of a nontrivial connected graph is a spanning tree in which every vertex has degree congruent to one modulo $\ell$. This notion provides a common generalization of classical spanning trees and odd spanning trees. We show, via a constructive greedy algorithm, that every $n$-vertex graph $G$ satisfying $n\equiv2\pmod{\ell}$ and $\delta(G)>\frac{(\ell-1)n}{\ell}$ has an $\ell$-congruent spanning tree. For the special case of odd spanning trees ($\ell=2$), our algorithmic approach simplifies the original proof by Zheng and Wu. We also derive formulas for the numbers of $\ell$-congruent spanning trees in complete graphs and complete bipartite graphs. These formulas specialize to the classical spanning-tree formulas when $\ell=1$ and to the corresponding odd-spanning-tree formulas when $\ell=2$.

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Zhidan Yan, Wei Wang. 2026-08-12. On spanning trees whose degrees are congruent to one modulo $\ell$. https://arxiv.org/abs/2608.11990

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