The domatic number of a graph is the maximum number of vertex disjoint dominating sets that partition the vertex set of the graph. In this paper we consider the fractional variant of this notion. Graphs with fractional domatic number 1 are exactly the graphs that contain an isolated vertex. Furthermore, it is known that all other graphs have fractional domatic number at least 2. In this note we characterize graphs with fractional domatic number 2. More specifically, we show that a graph without isolated vertices has fractional domatic number 2 if and only if it has a vertex of degree 1 or a connected component isomorphic to a 4-cycle. We conjecture that if the fractional domatic number is more than 2, then it is at least 7/3.

翻译：图的控制数（domatic number）是指将顶点集划分为顶点不相交的支配集的最大数目。本文考虑该概念的分数变体。分数控制数为1的图恰好是包含孤立顶点的图。此外，已知所有其他图的分数控制数至少为2。本文刻画了分数控制数为2的图。具体而言，我们证明：无孤立顶点的图其分数控制数为2当且仅当它包含一个度为1的顶点或一个同构于4-圈（4-cycle）的连通分支。我们猜想：若分数控制数大于2，则其值至少为7/3。