Sub-problems of the (3, 14) cage problem and their computer analysis

TL;DR

This paper investigates the (3, 14) cage problem by decomposing it into sub-problems based on symmetry factors, using computer searches to find minimal graphs, and providing evidence supporting the current record graph's optimality.

Contribution

It introduces a symmetry factor parameter, decomposes the problem into sub-problems, and uses computational methods to analyze the existence of minimal (3, g) bipartite graphs.

Findings

Minimum order for various symmetry factors identified

Non-existence of certain (3, 14) graphs between bounds

Supports current record graph as the (3, 14) cage

Abstract

A (k, g) graph is a graph with regular degree k and girth g. The cage problem refers to finding the smallest (k, g) graph. The (3, 14) cage problem is known to be unresolved. In 2002, Exoo found a (3, 14) record graph with order 384. The trivalent cage problem is restricted in this paper to the Hamiltonian bipartite class of trivalent graphs. A parameter called symmetry factor for representing rotational symmetry is introduced in this paper. The general problem of finding a (3, g) Hamiltonian bipartite graph of minimum order is further decomposed into a set of sub-problems for finding (3, g) Hamiltonian bipartite graphs of minimum order for various symmetry factors. The minimum order for (3, g) Hamiltonian bipartite graphs for various symmetry factors has been found using computer search. This information about sub-problems also yields useful information about non-existence of (3, 14) Hamiltonian bipartite graphs between the (3, 14) lower bound, 258 and the (3, 14) upper bound, 384. This non-existence information partially supports the likelihood of the current (3, 14) record graph indeed being the (3, 14) cage.