On the fractional total domatic numbers of incidence graphs

For a hypergraph $H$ with vertex set $X$ and edge set $Y$, the incidence graph of hypergraph $H$ is a bipartite graph $I(H)=(X,Y,E)$, where $xy\in E$ if and only if $x\in X$, $y\in Y$ and $x\in y$. A total dominating set of graph $G$ is a vertex subset that intersects every open neighborhood of $G$. Let $\mathcal{M}$ be a family of (not necessarily distinct) total dominating sets of $G$ and $r_{\mathcal{M}}$ be the maximum times that any vertex of $G$ appears in $\mathcal{M}$. The fractional domatic number $G$ is defined as $FTD(G)=\underset{\mathcal{M}}{sup}\frac{|\mathcal{M}|}{r_{\mathcal{M}}}$. In 2018, Goddard and Henning showed that the incidence graph of every complete $k$-uniform hypergraph $H$ with order $n$ has $FTD(I(H))=\frac{n}{n-k+1}$ when $n\ge 2k\ge 4$. We extend the result to the range $n>k\ge 2$. More generally, we prove that every balanced $n$-partite complete $k$-uniform hypergraph $H$ has $FTD(I(H))=\frac{n}{n-k+1}$ when $n\ge k$ and $H≆K_n^{(n)}$, where $FTD(I(K_n^{(n)}))=1$.