  Euler graphs are characterized by the simple criterion that degree of each
node is even. By restricting on the cycle types yet additional intrinsic
properties of Euler graphs are unveiled. For example, regularity higher than
degree two is impossible within the class {\epsilon}i of Euler graphs with one
type of cycles Cn, n=i(mod 4), i=0,1,2,3. Further, graphs in {\epsilon}i are
planar for i=1,2,3. In the light of new properties of Euler graphs more
gracefulness boundaries are conjectured for subclasses of Euler graphs and
where relevant extended for general class of graphs. In absence of general
analytical results much of the published papers resort to proving an infinite
class of graphs graceful or nongraceful. The purpose of this paper is not to
give families of graphs graceful or not. Instead, based on the available
information expected gracefulness boundaries are proposed which may guide where
to look for graceful graphs or lead to characterizations. While the
(Ringel,Kotzig,Rosa) Tree Conjecture continues to remain unsettled, the work
reported here serves an update on the conjectures made in Rao Hebbare
(1975,1981) and more conjectures subsequently made in Rao (1999,2000) based on
embedding theorems and graceful algorithms for constructing graceful graphs
from a graceful graph. It is hoped that these conjectures lead to analytical
techniques for establishing gracefulness property. Further probe into Euler
graphs with only two types of cycles and other combinations of cycles
continues.
