Independence and Matching Number in Graphs with Maximum Degree 4

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

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