Computational determination of (3,11) and (4,7) cages

TL;DR

This paper determines the minimal order and uniqueness of certain (k,g)-cages, providing new bounds and examples through heuristic and backtracking methods.

Contribution

It proves the minimality and uniqueness of the (3,11)-cage of order 112, finds a (4,7)-cage of order 67, and improves bounds for (3,13) and (3,14)-cages.

Findings

(3,11)-cage of order 112 is minimal and unique

(4,7)-cage of order 67 found

Lower bounds for (3,13) and (3,14)-cages improved

Abstract

A (k,g)-graph is a k-regular graph of girth g, and a (k,g)-cage is a (k,g)-graph of minimum order. We show that a (3,11)-graph of order 112 found by Balaban in 1973 is minimal and unique. We also show that the order of a (4,7)-cage is 67 and find one example. Finally, we improve the lower bounds on the orders of (3,13)-cages and (3,14)-cages to 202 and 260, respectively. The methods used were a combination of heuristic hill-climbing and an innovative backtrack search.