Levelness versus almost Gorensteinness of edge rings of complete multipartite graphs

Levelness and almost Gorensteinness are well-studied properties on graded rings as a generalized notion of Gorensteinness. In the present paper, we study those properties for the edge rings of the complete multipartite graphs, denoted by k[K-r1,..,r(n)] with 1 <= r(1) <= . . . <= r(n). We give the complete characterization of which k[K-r1,...,r(n)] is level in terms of n and r(1),...,r(n) Similarly, we also give the complete characterization of which k[K-r1,...,r(n)] is almost Gorenstein in terms of n and r(1),...,r(n).