  We prove that, for every integer $d$ with $d\geq 3$, there is an
approximation algorithm for the maximum induced matching problem restricted to
$\{ C_3,C_5\}$-free $d$-regular graphs with performance ratio
$0.708\bar{3}d+0.425$, which answers a question posed by Dabrowski et al.
(Theor. Comput. Sci. 478 (2013) 33-40). Furthermore, we show that every graph
with $m$ edges that is $k$-degenerate and of maximum degree at most $d$ with
$k<d$, has an induced matching with at least $m/((3k-1)d-k(k+1)+1)$ edges.
