Structural Characterisations of (n-1,n)-Trees

TL;DR

This paper develops a comprehensive higher-dimensional theory of (n-1,n)-trees in simplicial complexes, extending classical graph tree properties and correcting previous conjectures with explicit counterexamples.

Contribution

It introduces refined notions of paths and circuits for (n-1,n)-trees, establishes their characterizations, and disproves and corrects earlier conjectures in the field.

Findings

Established higher-dimensional analogues of classical tree properties.

Constructed explicit counterexamples to Dewdney's conjectures.

Formulated corrected conjectures with additional conditions.

Abstract

We study higher-dimensional analogues of graph-theoretic trees within the class of pure n-simplicial complexes. Focusing on the case m = n-1 in Dewdney's (m, n)-tree framework, we introduce refined notions of path and circuit sequences that overcome the structural limitations of existing definitions. Using these refinements, we establish higher-dimensional analogues of the classical characterisations of trees in graphs, including equivalences based on connectivity, acyclicity, path uniqueness, and enumerative constraints. We further disprove two conjectures posed by Dewdney by constructing explicit counterexamples, and we formulate corrected versions that hold under additional necessary conditions in the case m = n-1. These results provide a structurally complete theory of (n-1, n)-trees, parallel to the classical theory of graph-theoretic trees.