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

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.