Maximal induced matchings in K4-free and K5-free graphs

An induced matching in a graph is a set of edges whose endpoints induce a 1-regular subgraph. It is known that every graph on $n$ vertices has at most $10^{n/5}\approx 1.5849^n$ maximal induced matchings, and this bound is the best possible as any disjoint union of complete graphs $K_5$ forms an extremal graph. It is also known that the bound drops to $3^{n/3}\approx 1.4423^n$ when the graphs are restricted to the class of triangle-free ($K_3$-free) graphs. The extremal graphs, in this case, are known to be the disjoint unions of copies of $K_{3,3}$. Along the same line, we study the maximum number of maximal induced matchings when the graphs are restricted to $K_5$-free graphs and $K_4$-free graphs. We show that every $K_5$-free graph on $n$ vertices has at most $6^{n/4}\approx 1.5651^n$ maximal induced matchings and the bound is the best possible obtained by any disjoint union of copies of $K_4$. When the graphs are restricted to $K_4$-free graphs, the upper bound drops to $8^{n/5}\approx 1.5158^n$, and it is achieved by the disjoint union of copies of the wheel graph $W_5$.