Fractional forcing number of graphs

TL;DR

This paper introduces a continuous analogue of forcing sets for perfect matchings called forcing functions, extending the concept to fractional perfect matchings, and applies this to derive new bounds for regular graphs, including hypercubes.

Contribution

It defines the forcing function for fractional perfect matchings, establishing its properties and using it to obtain new bounds on forcing numbers in regular edge-transitive graphs.

Findings

Forcing function is a continuous, concave extension of forcing sets.

New upper bounds for maximum forcing numbers of hypercube graphs.

Results connect fractional and integral forcing concepts, improving bounds.

Abstract

The notion of forcing sets for perfect matchings was introduced by Harary, Klein, and \v{Z}ivkovi\'{c}. The application of this problem in chemistry, as well as its interesting theoretical aspects, made this subject very active. In this work, we introduce the notion of forcing function of fractional perfect matchings, which is continuous analogous to forcing sets defined over the perfect matching polytope of graphs. We show that this object is a continuous and concave function extension of the integral forcing set. Then, we use our results in the continuous world to conclude new bounds and results in the discrete case of forcing sets, for the family of regular edge-transitive graphs. In particular, we derive new upper bounds for the maximum forcing number of hypercube graphs.