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