Graphs of kei and their diameters

A kei on $[n]$ can be thought of as a set of maps $(f_x)_{x \in [n]}$, where each $f_x$ is an involution on $[n]$ such that $(x)f_x = x$ for all $x$ and $f_{(x)f_y} = f_yf_xf_y$ for all $x$ and $y$. We can think of kei as loopless, edge-coloured multigraphs on $[n]$ where we have an edge of colour $y$ between $x$ and $z$ if and only if $(x)f_y = z$; in this paper we show that any component of diameter $d$ in such a graph must have at least $2^d$ vertices and contain at least $2^{d-1}$ edges of the same colour. We also show that these bounds are tight for each value of $d$.