Fractional Disjoint Transversal Numbers And Fractional Total Domatic Numbers On Hypergraphs

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 if and only if x ∈ X,y ∈ Y and x ∈ 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 total domatic number G is defined as FTD(G)=(?).Let F be a family of(not necessarily distinct)transversals of hypergraph H and rF be the maximum times that any vertex of H appears in F.The fractional disjoint transversal number H is defined as DT_f(H)=(?).This paper studies the problems on the fractional disjoint transversal numbers of several classes of hypergraphs and the fractional total domatic numbers of the incidence graphs of hypergraphs.We determine the fractional disjoint transversal numbers of n-partite complete k-uniform hypergraphs,i.e.n/(n-k+1),and give the asymptotic value m of the fractional disjoint transversal numbers of(2,m)-degenerate planes.For all integers n and k no less than 2,we determine the fractional total domatic numbers of the incidence graphs of the balanced n-partite complete k-uniform hypergraph,i.e.n/(n-k+1),which generalize the relevant results of Goddard and Henning in 2018 on the fractional total domatic number of the complete k-uniform hypergraph.Moreover,the fractional total domatic number of the incidence graph of finite projective plane is determined.