Large Cayley digraphs and bipartite Cayley digraphs of odd diameters

TL;DR

This paper constructs large Cayley and bipartite Cayley digraphs with odd diameters, providing improved bounds on their maximum sizes for given degree and diameter using novel group automorphism techniques.

Contribution

The authors introduce new constructions for Cayley and bipartite Cayley digraphs that improve known bounds for odd diameters, employing innovative methods based on group automorphisms.

Findings

Constructed Cayley digraphs with order at least 2k(floor(d/2))^k for odd k

Constructed bipartite Cayley digraphs with order at least 2(k-1)(floor(d/2))^{k-1} for odd k

Provided the best known bounds on the sizes of such digraphs for large degree and odd diameter

Abstract

Let $C_{d,k}$ be the largest number of vertices in a Cayley digraph of degree $d$ and diameter $k$, and let $BC_{d,k}$ be the largest order of a bipartite Cayley digraph for given $d$ and $k$. For every degree $d\geq2$ and for every odd $k$ we construct Cayley digraphs of order $2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ and diameter at most $k$, where $k\ge 3$, and bipartite Cayley digraphs of order $2(k-1)\left(\lfloor\frac{d}{2}\rfloor\right)^{k-1}$ and diameter at most $k$, where $k\ge 5$. These constructions yield the bounds $C_{d,k} \ge 2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ for odd $k\ge 3$ and $d\ge \frac{3^{k}}{2k}+1$, and $BC_{d,k} \ge 2(k-1)\left(\lfloor\frac{d}{2}\rfloor\right)^{k-1}$ for odd $k\ge 5$ and $d\ge \frac{3^{k-1}}{k-1}+1$. Our constructions give the best currently known bounds on the orders of large Cayley digraphs and bipartite Cayley digraphs of given degree and odd diameter $k\ge 5$. In our proofs we use new techniques based on properties of group automorphisms of direct products of abelian groups.