Corrigendum to "Topology of matching complexes of complete graphs via discrete Morse theory'' [arXiv:2305.02973, Discrete Math. Theor. Comput. Sci. 26:3#13 (2024)]

TL;DR

This corrigendum corrects a mistaken motivation in a previous paper, disproves a conjecture about matching complexes of complete graphs, and establishes an optimal gradient vector field for the matching complex of K7.

Contribution

It provides a corrected theoretical foundation, disproves an earlier conjecture, and constructs an optimal gradient vector field for a specific matching complex.

Findings

Disproves Conjecture 5.1 about matching complexes.

Establishes an optimal gradient vector field on the matching complex of K7.

Clarifies the motivation behind the original conjecture was mistaken.

Abstract

In this corrigendum to arXiv:2305.02973, we justify that the motivation behind Conjecture 5.1 is based on a mistaken notion. Moreover, we prove a stronger theorem that disproves the conjecture, and obtain an optimal gradient vector field on the matching complex of the complete graph of order 7.