The total Betti number of the independence complex of ternary graphs
Wentao Zhang, Hehui Wu · Journal of the European Mathematical Society · 2023
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.