  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.
