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arXiv · 2011.10939

The Betti Number of the Independence Complex of Ternary Graphs

Hehui Wu·Wentao Zhang
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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.

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Hehui Wu, Wentao Zhang. 2020-11-22. The Betti Number of the Independence Complex of Ternary Graphs. https://arxiv.org/abs/2011.10939

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