New Results Relating Independence and Matchings

In this paper we study relationships between the matching number, written mu(G), and the independence number, written alpha(G). Our first main result is to show alpha(G) <= mu(G) + |X| - mu(G[N-G[X]]), where X is any intersection of maximum independent sets in G. Our second main result is to show delta(G) alpha(G) <= 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.