Betti splittings and multigraded Betti numbers of cover ideals of bipartite graphs

In this note, we study Betti splittings of cover ideals of bipartite graphs. We prove that if $J\subset \mathbb{k}[x_1,\dots ,x_n]$ 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.