Inequalities for Total Matchings of Graphs

Abstract

As a generalization of a matching consisting of edges only, Alavi et al. in [1] define a total matching which may contain both edges and vertices. Using total matchings, [1] defines a parameter ${\beta }_2^{\prime }(G)$ and proves that ${\beta }_2^{\prime }(G)\le p-1$ holds for a connected graph of order $p\ge 2$.
Our main result is to improve this inequality to ${\beta }_2^{\prime }(G)\le p-2\sqrt{p}+2$ and we give an example demonstrating this bound to be best possible.
Relations of several other parameters to ${\beta }_2^{\prime }$ are demonstrated.