Spanning trees in directed square cycles
Yuuho Tanaka
TL;DR
Problem: count directed spanning trees in the directed square cycle $\overrightarrow{C_n^2}$ without using the Matrix Tree Theorem. Approach: classify nontrivial weakly connected spanning closed subgraphs as $\overrightarrow{S_{n,k,j}}$ and, via $C$-homotopy, show every spanning tree is contained in a unique such subgraph, reducing counting to sums over directed strip graphs. Key results: the total number rooted at any vertex satisfies $t(\overrightarrow{C_n^2},v_j)=J_n$, where $J_n$ is the Jacobsthal sequence; moreover, $t(\overrightarrow{S_{n,k,j}},v_j)=t(\overrightarrow{S_{n-2k}},1)$. Significance: provides a purely combinatorial proof of the counting formula, linking directed cycle powers to Jacobsthal numbers and suggesting extensions to higher powers.
Abstract
We classify weakly connected spanning closed (WCSC) subgraphs of $\overrightarrow{C_n^2}$, the square of a directed $n$-vertex cycle. Then we show that every spanning tree of $\overrightarrow{C_n^2}$ is contained in a unique nontrivial WCSC subgraph of $\overrightarrow{C_n^2}$. As a result, we obtain a purely combinatorial derivation of the formula for the number of directed spanning trees of $\overrightarrow{C_n^2}$. Moreover, we obtain the formula for the number of directed spanning trees of $\overrightarrow{C_n^2}$, which is a Jacobsthal number.