A note on maximum fractional matchings of graphs

A fractional matching of a graph G is a function f giving each edge a number in [0, 1] so that $$\sum _{e\in \Gamma _G (v)} f(e)\le 1 $$ for each $$v \in V(G)$$ , where $$\Gamma _G (v)$$ is the set of edges incident to v in G. The fractional matching number of G, denoted by $$\mu _f(G)$$ , is the maximum of $$\sum _{e\in E(G)} f(e)$$ over all fractional matchings f. A fractional matching f of G is called a maximum fractional matching if $$\sum _{e\in E(G)} f(e)=\mu _f(G)$$ . In this paper, as a supplement of known results in Liu et al. (J Comb Optim 40:59–68, 2020), we further study the maximum fractional matching, and give some sufficient and necessary conditions to characterize the maximum fractional matching. Furthermore, as applications, the sharp lower bounds of the fractional matching number for Cartesian product, strong product, lexicographic product and direct product are obtained.