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 addresses a mistaken motivation behind Conjecture 5.1 in the study of the matching complex $M_n$ of the complete graph via discrete Morse theory. It first exposes the erroneous premise about gradient-path correspondences and provides a concrete counterexample using the critical $2$-simplex $\eta_3$ to disprove the conjecture. It then proves a stronger result: by three simultaneous gradient-path cancellations, one can obtain an optimal gradient vector field $\mathcal{M}^\odot$ on $M_7$ with $21$ critical $2$-simplices, $1$ critical $1$-simplex, and $1$ critical $0$-simplex, matching the theoretical lower bounds. This clarifies the structure of the Morse decomposition for $M_7$ and strengthens the methodological toolkit for analyzing matching complexes via discrete Morse theory.

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.

Figures (3)

• Figure 1: All three possible $\mathcal{M}^*$-paths that start from a $1$-simplex contained in $\eta_3$, and end at a critical $1$-simplex.
• Figure 2: Only two possible $\mathcal{M}^*$-paths that start from a $1$-simplex contained in $\eta_1$, and end at a critical $1$-simplex.
• Figure 3: Only two possible $\mathcal{M}^*$-paths that start from a $1$-simplex contained in $\eta_2$, and end at a critical $1$-simplex.

Theorems & Definitions (2)

• Theorem 2.1
• proof