The difference and ratio of the fractional matching number and the matching number of graphs

Abstract

Given a graph G, the matching number of G, written alpha'(G), is the maximum size of a matching in G, and the fractional matching number of G, written alpha(f)'(G), is the maximum size of a fractional matching of G. In this paper, we prove that if G is an n-vertex connected graph that is neither K-1 nor K-3, then alpha(f)' (G)- alpha'(G) <= n-2/6 and alpha(f)'(G)/alpha(f)'(G) <= 3n/2n+2. Both inequalities are sharp, and we characterize the infinite family of graphs where equalities hold.