Convex subgraphs and spanning trees of the square cycles

TL;DR

This paper classifies convex subgraphs of square cycles and demonstrates a unique containment property of spanning trees within these subgraphs, providing a combinatorial derivation for counting spanning trees.

Contribution

It introduces a classification of convex subgraphs of square cycles and establishes a unique containment relation for spanning trees, leading to a new combinatorial formula.

Findings

Classified connected spanning convex subgraphs of square cycles

Proved each spanning tree is contained in a unique convex subgraph

Derived a combinatorial formula for the number of spanning trees

Abstract

We classify connected spanning convex subgraphs of the square cycles. We then show that every spanning tree of $C_n^2$ is contained in a unique nontrivial connected spanning convex subgraph of $C_n^2$. As a result, we obtain a purely combinatorial derivation of the formula for the number of spanning trees of the square cycles.