  The subclass of Euler graphs with only one type of cycles under (mod 4)
operation was studied in Part-1 of this series. It was established that such
graphs under regularity are nonexistent for degree >2. Here we consider the
subclass of Euler graphs with only two types of cycles under (mod 4) operation.
Six cases arise. The case when the cycle types are 0&2(mod 4), the well known
class of bipartite Euler graphs, the existence of regular bipartite Euler
graphs is very much known. In the other five cases, it transpires that regular
Euler graphs with only two types of cycles are nonexistent. Some constructions
of Euler graphs with the property are given. We investigate some properties of
cycle decompositions, block structure and cycle intersections. Cycle
decomposition of Euler graphs allows segregating Euler graphs satisfying
Rosa-Golumb criterion and so are nongraceful. In the other case the graphs are
possible candidates for gracefulness and are conjectured graceful leading to
better understanding of gracefulness boundaries. The cases when the cycle types
are 1&2, 1&3, 2&3(mod 4) the graphs are proved to be planar and in other three
cases the graphs may not be planar with examples of nonplanar graphs. Probe
into the properties of Euler graphs is to propose expected gracefulness
boundaries which may guide where to look for graceful graphs. Further, the
conjectures may lead to analytical techniques for establishing gracefulness
property.
