@article{Zhang2023, 
author = {Yameng Zhang and Xia Zhang},
title = {On the fractional total domatic numbers of incidence graphs},
year = {2023},
journal = {Mathematical Modelling and Control},
volume = {3},
number = {1},
pages = {73-79},
keywords = {fractional, hypergraph, total dominating set, incident graph},
url = {https://www.sciopen.com/article/10.3934/mmc.2023007},
doi = {10.3934/mmc.2023007},
abstract = {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    x  y  ∈  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        r                  M             be the maximum times that any vertex of    G appears in        M  . The fractional domatic number    G is defined as    F  T  D  (  G  )  =      sup                  M                                |                    M                    |                    r                        M                    . In 2018, Goddard and Henning showed that the incidence graph of every complete    k-uniform hypergraph    H with order    n has    F  T  D  (  I  (  H  )  )  =      n          n      −      k      +      1       when    n  ≥  2  k  ≥  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    F  T  D  (  I  (  H  )  )  =      n          n      −      k      +      1       when    n  ≥  k and    H  ≆      K    n          (      n      )      , where    F  T  D  (  I  (      K    n          (      n      )        )  )  =  1.}
}