Narrow Results

In this paper, we study Betti splittings of cover ideals of bipartite graphs. We prove that if $J\subset 𝕜[x_1,\dots ,x_n]$ is the cover ideal of a bipartite graph then the $x_i$-partition of $J$ is a Betti splitting for any $i$. We also prove that multigraded Betti numbers of any squarefree monomial ideal can appear in a certain part of multigraded Betti numbers of the cover ideal of a bipartite graph.

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

We prove that the depth functions of cover ideals of balanced hypergraph have the nonincreasing property. Furthermore, we also give a bound for the index of depth stability of these ideals.