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

TL;DR

This paper investigates the relationship between the fractional matching number and the matching number in connected graphs, establishing sharp bounds and characterizing cases of equality.

Contribution

It proves tight bounds on the difference and ratio of fractional and integral matching numbers for connected graphs, extending understanding of their relationship.

Findings

Bound on the difference: '_f(G) - \u0017'(G)  (n-2)/6

Bound on the ratio: '_f(G)/'(G)  3n/(2n+2)

Characterization of graphs where bounds are tight

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 $\alpha'_f(G)-\alpha'(G) \le \frac{n-2}6$ and $\frac{\alpha'_f(G)}{\alpha'(G)} \le \frac{3n}{2n+2}$. Both inequalities are sharp, and we characterize the infinite family of graphs where equalities hold.