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 α′(G), is the maximum size of a matching in G, and the fractional matching number of G, written αf′(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 K1 nor K3, then αf′(G)−α′(G)≤n−26 and αf′(G)α′(G)≤3n2n+2. Both inequalities are sharp, and we characterize the infinite family of graphs where equalities hold.