Graphs of kei and their diameters

Abstract

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 $f_{(x)f_y}=f_y{f}_x{f}_y$ 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 $d$.

9 pages, 5 figures

References in corpus (1)

• The origins of involutory quandles