Nordhaus-Gaddum type inequality for the fractional matching number of a graph

TL;DR

This paper establishes sharp lower bounds for the sum of a graph's fractional matching number and that of its complement, characterizing extremal graphs for various conditions.

Contribution

It introduces new Nordhaus-Gaddum type inequalities for the fractional matching number and characterizes all extremal graphs achieving these bounds.

Findings

Sum of fractional matching numbers of G and its complement is at least n/2 for n ≥ 2.

Sum is at least (n+1)/2 for non-empty G and complement, with n ≥ 28.

Sum is at least (n+4)/2 when G and its complement have no isolated vertices, with n ≥ 28.

Abstract

The fractional matching number of a graph G, is the maximum size of a fractional matching of G. The following sharp lower bounds for a graph G of order n are proved, and all extremal graphs are characterized in this paper. (1)The sum of the fractional matching number of a graph G and the fractional matching number of its complement is not less than n/2 , where n is not less than 2. (2) If G and its complement are non-empty, then the sum of the fractional matching number of a graph G and the fractional matching number of its complement is not less than (n+1)/2, where n is not less than 28. (3) If G and its complement have no isolated vertices, then the sum of the fractional matching number of a graph G and the fractional matching number of its complement is not less than (n+4)/2, where n is not less than 28.