  Let $\mathbb{K}$ be a field and $R = \mathbb{K}[x_1, \ldots, x_n]$. We obtain
an improved upper bound for asymptotic resurgence of squarefree monomial ideals
in $R$. We study the effect on the resurgence when sum, product and
intersection of ideals are taken. We obtain sharp upper and lower bounds for
the resurgence and asymptotic resurgence of cover ideals of finite simple
graphs in terms of associated combinatorial invariants. We also explicitly
compute the resurgence and asymptotic resurgence of cover ideals of several
classes of graphs. We characterize a graph being bipartite in terms of the
resurgence and asymptotic resurgence of edge and cover ideals. We also compute
explicitly the resurgence and asymptotic resurgence of edge ideals of some
classes of graphs.
