# The total Betti number of the independence complex of ternary graphs

### Wentao Zhang

Fudan University, Shanghai, China### Hehui Wu

Fudan University, Shanghai, China

## Abstract

Given a graph $G$, the *independence complex* $I(G)$ is the simplicial complex whose faces are the independent sets of $V(G)$. Let $b~_{i}$ denote the $i$-th reduced Betti number of $I(G)$, and let $b(G)$ denote the sum of the $b~_{i}(G)$’s. A graph is ternary if it does not contain induced cycles with length divisible by 3. Kalai and Meshulam conjectured that $b(G)≤1$ whenever $G$ is ternary. We prove this conjecture. This extends a recent result proved by Chudnovsky, Scott, Seymour and Spirkl that for any ternary graph $G$, the number of independent sets with even cardinality and the number of independent sets with odd cardinality differ by at most 1.

## Cite this article

Wentao Zhang, Hehui Wu, The total Betti number of the independence complex of ternary graphs. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1378