Independence in graphs with maximum degree four

Let $\text{C}$ be the class of triangle-free graphs with maximum degree four. A lower bound for the number of edges in a graph of $\text{C}$ is derived in terms of its order p and independence β. Also a characterization of certain minimum independence graphs in $\text{C}$ is provided. Let r(k) be the smallest integer such that every graph in $\text{C}$ with at least r(k) vertices has independence at least k. The values of r(k) for all k may be derived from our main theorem and $\text{4}\text{13}$ obtained as the best possible lower bound for the independence ratio $\text{β}\text{p}$ of graphs in $\text{C}$.