  We prove that $\frac{7}{4}\alpha(G)+\beta(G)\geq n(G)$ and
$\alpha(G)+\frac{3}{2}\beta(G)\geq n(G)$ for every triangle-free graph $G$ with
maximum degree at most $4$, where $\alpha(G)$ is the independence number and
$\beta(G)$ is the matching number of $G$, respectively. These results are sharp
for a graph on $13$ vertices. Furthermore we show $\chi(G)\leq
\frac{7}{4}\omega(G)$ for $\{3K_1,K_1\cup K_5\}$-free graphs, where $\chi(G)$
is the chromatic number and $\omega(G)$ is the clique number of $G$,
respectively.

Combinatorics