New results relating independence and matchings

TL;DR

This paper establishes new bounds relating the matching number and independence number of a graph, generalizing previous relations and providing examples to demonstrate these improvements.

Contribution

The paper introduces novel bounds connecting independence and matching numbers, extending known relations and applying to any intersection of maximum independent sets.

Findings

Derived an upper bound for independence number involving maximum independent set intersections.

Established a degree-based inequality linking independence and matching numbers.

Provided examples illustrating the improvements over previous bounds.

Abstract

In this paper we study relationships between the \emph{matching number}, written $\mu(G)$, and the \emph{independence number}, written $\alpha(G)$. Our first main result is to show \[ \alpha(G) \le \mu(G) + |X| - \mu(G[N_G[X]]), \] where $X$ is \emph{any} intersection of maximum independent sets in $G$. Our second main result is to show \[ \delta(G)\alpha(G) \le \Delta(G)\mu(G), \] where $\delta(G)$ and $\Delta(G)$ denote the minimum and maximum vertex degrees of $G$, respectively. These results improve on and generalize known relations between $\mu(G)$ and $\alpha(G)$. Further, we also give examples showing these improvements.