Year: 2026
Type: preprint
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 $δ(G)>\frac{(\ell-1)n}{\ell}$ 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 $\ell=2$.
Source: arXiv (Cornell University)
Authors: Yan, Zhidan, Wang, Wei
Institution: Anhui Polytechnic University
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Award: G3496664386
OpenAlex ID: W7202361083
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