Betti numbers of chordal graphs and $f$-vectors of simplicial complexes

Abstract

Let $G$ be a chordal graph and $I(G)$ its edge ideal. Let $\beta (I(G))=({\beta }_0,{\beta }_1,...,{\beta }_p)$ denote the Betti sequence of $I(G)$, where ${\beta }_i$ stands for the $i$th total Betti number of $I(G)$ and where $p$ is the projective dimension of $I(G)$. It will be shown that there exists a simplicial complex $\Delta$ of dimension $p$ whose $f$-vector $f(\Delta )=(f_0,f_1,...,f_p)$ coincides with $\beta (I(G))$.

15 pages