三元图的独立复形的贝蒂数
The Betti Number of the Independence Complex of Ternary Graphs
摘要：
给定一个图$G$，\textit{independence complex} $I(G)$是单纯复形，其面是$V(G)$的独立集。令$\tilde{b}_i$表示$I(G)$的第$i$次约化贝蒂数，令$b(G)$表示$\tilde{b}_i(G)$的和.一个图是三元的，如果它不包含长度可被3整除的诱导圈。G. Kalai和K. Meshulam指出，只要$G$是三进制，就应使用$b(G)\le 1$。我们证明了这个猜想。这推广了Chudnovsky，Scott，Seymour和Spirkl最近的一个结果：对于任意三元图$G$，具有偶数基数的独立集和具有奇数基数的独立集的个数相差至多1.
英文摘要：
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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