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# Mathematics > Combinatorics

# Title: The Betti Number of the Independence Complex of Ternary Graphs

(Submitted on 22 Nov 2020 (v1), last revised 18 Oct 2021 (this version, v3))

Abstract: Given a graph $G$, the \textit{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 $\tilde{b}_i(G)$'s. A graph is ternary if it does not contain induced cycles with length divisible by three. G. Kalai and K. Meshulam conjectured that $b(G)\le 1$ whenever $G$ is ternary. We prove this conjecture. This extends a recent results proved by Chudnovsky, Scott, Seymour and Spirkl that for any ternary graph $G$, the number of independent sets with even cardinality and the independent sets with odd cardinality differ by at most 1.

## Submission history

From: Hehui Wu [view email]**[v1]**Sun, 22 Nov 2020 05:25:24 GMT (9kb)

**[v2]**Mon, 17 May 2021 18:07:40 GMT (10kb)

**[v3]**Mon, 18 Oct 2021 11:20:55 GMT (10kb)

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