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

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 α 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 α f ' ( G ) - α ' ( G ) ? n - 2 6 and α f ' ( G ) α ' ( G ) ? 3 n 2 n + 2 . Both inequalities are sharp, and we characterize the infinite family of graphs where equalities hold.