For an integer \(r \geq 2\), an \(r\)-uniform hypergraph is a pair \(H=(V(H),E(H))\) with vertex set \(V(H)\) and edge set \(E(H) \subseteq \genfrac{(}{)}{0pt}{1}{V(H)}{r}\). The matching number of \(H\) is the size of a maximum matching in \(H\).
A fractional matching of an \(r\)-uniform hypergraph \(H\) is a function \(f\) assigning each edge with a real number in \([0, 1]\) so that \( \sum_{e \in \Gamma(v)} {f(e)} \leq 1\) for each \(v \in V (H)\), where \(\Gamma(v)\) is the set of edges incident with \(v\) in \(H\). The fractional matching number of \(H\) is the maximum value of \(\sum_{e \in E(H)} {f(e)} \) over all fractional matchings \(f\).
In this paper, for three integers \(n\), \(k\), and \(d\), the authors determine the maximum size of a graph on \(n\) vertices with fractional matching number \(k\) and maximum degree at most \(d\). As a consequence, they obtain the maximum size of a graph with a given number of vertices and a fractional matching number. This partially confirms a conjecture proposed by N. Alon et al. [J. Comb. Theory, Ser. A 119, No. 6, 1200–1215 (2012; Zbl 1242.05189)] on the maximum size of \(r\)-uniform hypergraph with a fractional matching number for the special case when \(r = 2\).