The Merrifield-Simmons conjecture holds for bipartite graphs

TL;DR

This paper proves the Merrifield-Simmons conjecture for bipartite graphs by analyzing the sign of a specific term related to independent set counts, extending previous results to a generalized vertex deletion context.

Contribution

The paper establishes the validity of the Merrifield-Simmons conjecture for bipartite graphs using a generalized approach involving vertex subset deletions.

Findings

The conjecture holds for bipartite graphs.

Sign of the term depends on the parity of the distance between vertices.

Generalization to vertex subsets extends previous results.

Abstract

Let $G = (V, E)$ be a graph and $\sigma(G)$ the number of independent (vertex) sets in $G$. Then the Merrifield-Simmons conjecture states that the sign of the term $\sigma(G_{-u}) \cdot \sigma(G_{-v}) - \sigma(G) \cdot \sigma(G_{-u-v})$ only depends on the parity of the distance of the vertices $u, v \in V$ in $G$. We prove that the conjecture holds for bipartite graphs by considering a generalization of the term, where vertex subsets instead of vertices are deleted.