A new approach to catalog small graphs of high even girth

TL;DR

This paper introduces an infinite catalog of (3,g) graphs with even girth g, focusing on Hamiltonian bipartite graphs, providing new minimal graphs, a compact notation, and insights into symmetry and graph existence.

Contribution

It presents a novel infinite catalog of (3,g) graphs for even girth g, using Hamiltonian bipartite graphs and a new notation, expanding known graph listings and analysis methods.

Findings

Identified Hamiltonian bipartite graphs as a promising class for (3,g) graphs.

Catalog contains many non-vertex-transitive graphs.

Introduced D3 chord index notation for concise graph representation.

Abstract

A catalog of a class of (3,g) graphs for even girth g is introduced in this paper. A (k,g) graph is a regular graph with degree k and girth g. This catalog of (3,g) graphs for even girth g satisfying 6 <= g <= 16, has the following properties. Firstly, this catalog contains the smallest known (3, g) graphs. An appropriate class of cubic graphs for this catalog has been identified, such that the (3,g) graph of minimum order within the class is also the smallest known (3,g) graph. Secondly, this catalog contains (3,g) graphs for more orders than other listings. Thirdly, the class of graphs have been defined so that a practical algorithm to generate graphs can be created. Fourthly, this catalog is infinite, since the results are extended into knowledge about infinitely many graphs. The findings are as follows. Firstly, Hamiltonian bipartite graphs have been identified as a promising class of cubic graphs that can lead to a catalog of (3,g) graphs for even girth g with graphs for more orders than other listings, that is also expected to contain a (3,g) graph with minimum order. Secondly, this catalog of (3,g) graphs contains many non-vertex-transitive graphs. Thirdly, in order to make the computation more tractable, and at the same time, to enable deeper analysis on the results, symmetry factor has been introduced as a measure of the extent of rotational symmetry along the identified Hamiltonian cycle. The D3 chord index notation is introduced as a concise notation for cubic Hamiltonian bipartite graphs. The D3 chord index notation is twice as compact as the LCF notation. The D3 chord index notation can specify an infinite family of graphs. Fourthly, results on the minimum order for existence of a (3,g) Hamiltonian bipartite graph, and minimum value of symmetry factor for existence of a (3,g) Hamiltonian bipartite graph are of wider interest.