New small regular graphs of given girth: the cage problem and beyond

TL;DR

This paper develops computational algorithms to find small regular graphs with specified girth, improving known bounds for several cage problem cases and addressing variants to narrow the bounds gap.

Contribution

It introduces four new graph generation algorithms and achieves new upper bounds for multiple cage problem cases, some improving bounds established over two decades.

Findings

Established new upper bounds for 11 cage problem cases.

Improved the bound for n(4,10) from 384 to 320.

Adapted algorithms for cage variants, further narrowing bounds.

Abstract

The cage problem concerns finding $(k,g)$-graphs, which are $k$-regular graphs with girth $g$, of the smallest possible number of vertices. The central goal is to determine $n(k,g)$, the minimum order of such a graph, and to identify corresponding extremal graphs. In this paper, we study the cage problem and several of its variants from a computational perspective. Four complementary graph generation algorithms are developed based on exhaustive generation of lifts, a tabu search heuristic, a hill climbing heuristic and excision techniques. Using these methods, we establish new upper bounds for eleven cases of the classical cage problem: $n(3,16) \leq 936$, $n(3,17) \leq 2048$, $n(4,9) \leq 270$, $n(4,10) \leq 320$, $n(4,11) \leq 713$, $n(5,9) \leq 1116$, $n(6,11) \leq 7783$, $n(8,7) \leq 774$, $n(10,7) \leq 1608$, $n(12,7) \leq 2890$ and $n(14,7) \leq 4716$. Notably, our results improve upon several of the best-known bounds, some of which have stood unchanged for 22 years. Moreover, the improvement for $n(4,10)$, from the longstanding upper bound of 384 down to 320, is surprising and constitutes a substantial improvement. While the main focus is on the cage problem, we also adapted our algorithms for variants of the cage problem that received attention in the literature. For these variants, additional improvements are obtained, further narrowing the gaps between known lower and upper bounds.