The total Betti number of the independence complex of ternary graphs

Abstract

Given a graph $G$, the independence complex $I(G)$ is the simplicial complex whose faces are the independent sets of $V(G)$. Let $\tilde{b}\_i$ denote the $i$-th reduced Betti number of $I(G)$, and let $b(G)$ denote the sum of the $\tilde{b}\_i(G)$’s. A graph is ternary if it does not contain induced cycles with length divisible by 3. Kalai and Meshulam conjectured that $b(G)\le 1$ 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.