Proof of the Kalai-Meshulam conjecture

TL;DR

This paper proves the Kalai-Meshulam conjecture, establishing that for graphs with no induced cycles of length divisible by three, the sum over stable sets of (-1) to the size of the set is at most 1 in absolute value.

Contribution

The paper provides a proof of the Kalai-Meshulam conjecture, linking topological properties of graphs to combinatorial invariants.

Findings

Confirmed the conjecture that |f_G| ≤ 1 for graphs without induced cycles of length divisible by three.

Established the value of f_G as ±2 for cycles with length divisible by three.

Connected topological considerations with combinatorial graph properties.

Abstract

Let $G$ be a graph, and let $f_G$ be the sum of $(-1)^{|A|}$, over all stable sets $A$. If $G$ is a cycle with length divisible by three, then $f_G= \pm 2$. Motivated by topological considerations, G. Kalai and R. Meshulam made the conjecture that,if no induced cycle of a graph $G$ has length divisible by three, then $|f_G|\le 1$. We prove this conjecture.