Graphs of kei and their diameters

TL;DR

This paper investigates the structure of kei graphs, establishing tight bounds on the number of vertices and edges in components of a given diameter, revealing fundamental combinatorial properties.

Contribution

It introduces new bounds on the size and edge count of kei graph components based on their diameter, and proves these bounds are optimal.

Findings

Components of diameter d have at least 2^d vertices.

Such components contain at least 2^{d-1} edges of the same colour.

Bounds are tight for each diameter d.

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 $f_{(x)f_y} = f_yf_xf_y$ 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 $d$.