Induced Matchings in Graphs of Maximum Degree 4

For a graph $G$, let $ν_s(G)$ be the induced matching number of $G$. We prove the sharp bound $ν_s(G)\geq \frac{n(G)}{9}$ for every graph $G$ of maximum degree at most $4$ and without isolated vertices that does not contain a certain blown up $5$-cycle as a component. This result implies a consequence of the well known conjecture of Erdős and Nešetřil, saying that the strong chromatic index $χ_s'(G)$ of a graph $G$ is at most $\frac{5}{4}Δ(G)^2$, because $ν_s(G)\geq \frac{m(G)}{χ_s'(G)}$ and $n(G)\geq \frac{m(G)Δ(G)}{2}$. Furthermore, it is shown that there is polynomial-time algorithm that computes induced matchings of size at least $\frac{n(G)}{9}$.

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