Regularity of Powers of Bipartite Graphs

Abstract

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. For all $s\ge 1$, we obtain upper bounds for reg$(I(G{)}^s)$ for bipartite graphs. We then compare the properties of $G$ and $G^{\prime }$, where $G^{\prime }$ is the graph associated with the polarization of the ideal $(I(G{)}^{s+1}:e_1\cdots e_s)$, where $e_1,\dots e_s$ are edges of $G$. Using these results, we explicitly compute reg$(I(G{)}^s)$ for several subclasses of bipartite graphs.

19 pages