论文标题
三元图的独立复合物的贝蒂数
The Betti Number of the Independence Complex of Ternary Graphs
论文作者
论文摘要
给定图形$ g $，\ textit {独立综合体} $ i（g）$是简单的综合体，其面孔是$ v（g）$的独立集。令$ \ tilde {b} _i $表示$ i $ th减少$ i（g）$的betti数字，让$ b（g）$表示$ \ tilde {b} _i（g）$的总和。图形是三元，如果它不包含诱导的循环，长度可除以三个。 G. Kalai和K. Meshulam猜想$ b（g）\ le 1 $每当$ g $是三元时。我们证明了这个猜想。这扩展了Chudnovsky，Scott，Seymour和Spirkl的最新结果，即对于任何三元图$ G $，具有均匀基数的独立套件的数量和具有奇数奇数的独立套件的数量最多有1个。
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.
