On the order-diameter ratio of girth-diameter cages

TL;DR

This paper investigates the properties of girth-diameter cages, providing asymptotic bounds, exact values for certain cases, and an algorithm for generating and analyzing these graphs, including the largest known example.

Contribution

It offers new asymptotic bounds, exact calculations for specific parameters, and an exhaustive graph generation method for girth-diameter cages.

Findings

Asymptotic bounds for order-to-diameter ratio as diameter increases

Exact values and counts for (3;g,d)-cages with g in {4,5}

Generated and analyzed the largest known girth-diameter cage, a (3;7,35)-cage of order 136

Abstract

For integers $k,g,d$, a $(k;g,d)$-cage (or simply girth-diameter cage) is a smallest $k$-regular graph of girth $g$ and diameter $d$ (if it exists). The order of a $(k;g,d)$-cage is denoted by $n(k;g,d)$. We determine asymptotic lower and upper bounds for the ratio between the order and the diameter of girth-diameter cages as the diameter goes to infinity. We also prove that this ratio can be computed in constant time for fixed $k$ and $g$. We theoretically determine the exact values $n(3;g,d)$, and count the number of corresponding girth-diameter cages, for $g \in \{4,5\}$. Moreover, we design and implement an exhaustive graph generation algorithm and use it to determine the exact order of several open cases and obtain -- often exhaustive -- sets of the corresponding girth-diameter cages. The largest case we generated and settled with our algorithm is a $(3;7,35)$-cage of order 136.