  We will study monomial ideals $I$ in the exterior algebra as well as in the
polynomial ring whose generic initial ideal is constant for all term orders up
to permutations of variables. First, in the exterior algebra, we determine all
graphs and all flag complexes whose exterior face ideal satisfies the above
condition. Second, in the polynomial ring, it will be shown that the generic
initial ideal $\gin_\sigma(I(G))$ of the edge ideal $I(G)$ of a graph $G$ is
constant for all term orders $\sigma$ up to permutations of variables if and
only if $G$ is a complete bipartite graph.
