Some properties of catalog of (3, g) Hamiltonian bipartite graphs: orders, non-existence and infiniteness

TL;DR

This paper discusses a catalog of (3, g) bipartite graphs with even girth, highlighting its extensive coverage of graph orders, non-existence cases, and the fact that the catalog extends infinitely.

Contribution

It introduces a comprehensive catalog of (3, g) bipartite graphs that surpasses previous lists in the range of graph orders and proves the catalog's infinite extension.

Findings

Catalog includes more graph orders than previous lists.

Identifies specific orders and girths where graphs do not exist.

Proves the catalog extends infinitely.

Abstract

The focus of this paper is on discussion of a catalog of a class of (3, g) graphs for even girth g. A (k, g) graph is a graph with regular degree k and girth g. This catalog is compared with other known lists of (3, g) graphs such as the enumerations of trivalent symmetric graphs and enumerations of trivalent vertex-transitive graphs, to conclude that this catalog has graphs for more orders than these lists. This catalag also specifies a list of orders, rotational symmetry and girth for which the class of (3, g) graphs do not exist. It is also shown that this catalog of graphs extends infinitely.