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 $ \mathscr{M} $ be a family of (not necessarily distinct) total dominating sets of $ G $ and $ r_{\mathscr{M}} $ be the maximum times that any vertex of $ G $ appears in $ \mathscr{M} $. The fractional domatic number $ G $ is defined as $ FTD(G) = \sup_{\mathscr{M}}\frac{|\mathscr{M}|}{r_{\mathscr{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\geq 2k\geq 4 $. We extend the result to the range $ n > k\geq 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\geq k $ and $ H\ncong K_n^{(n)} $, where $ FTD(I(K_n^{(n)})) = 1 $.