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 $\mathbb{k}[K_{r_1,\dots ,r_n}]$ with $1\le r_1\le \cdots \le r_n$. We give the complete characterization of which $\mathbb{k}[K_{r_1,\dots ,r_n}]$ is level in terms of $n$ and $r_1,\dots ,r_n$. Similarly, we also give the complete characterization of which $\mathbb{k}[K_{r_1,\dots ,r_n}]$ is almost Gorenstein in terms of $n$ and $r_1,\dots ,r_n$.