AI 中文总结
本文研究度数模ℓ余1的生成树，通过构造性贪心算法证明满足特定条件的图存在此类生成树，简化了奇生成树的相关证明，并推导了完全图和完全二部图中该类生成树的数量公式。
AI 中文摘要
非平凡连通图的ℓ-同余生成树是指每个顶点度数模ℓ余1的生成树，该概念是经典生成树和奇生成树的统一推广。本文通过构造性贪心算法证明，每个满足n≡2(modℓ)且最小度δ(G)>(ℓ-1)n/ℓ的n顶点图G都存在ℓ-同余生成树；对于奇生成树（ℓ=2）的特殊情况，该算法方法简化了Zheng和Wu的原始证明。本文还推导了完全图和完全二部图中ℓ-同余生成树的数量公式，这些公式在ℓ=1时退化为经典生成树公式，在ℓ=2时退化为对应的奇生成树公式。
英文摘要
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 $\ell=2$.
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