  We propose a Law of Nature? Viz., Pure Regularity Occurs at Na\"ive Levels
and Regularity has Affinity with Evenness. In a series of three papers, it was
established that regular Euler graphs with only one type of (pure) cycles are
nonexistent; Regular Euler graphs with only two types of cycles are possible in
one of the six cases, viz., regular bipartite Euler graphs of degree >2;
Evenness plays role in unveiling regularity; Lastly, K5 is a regular Euler
graph with three types of cycles (0,1,3); This is the only known graph with the
property; It is conjectured that regular Euler graphs of order >5 with only
three cycle types are nonexistent and this is proved true in part cases in each
of the four cases. Some constructions and examples are given for the Euler
graphs under (mod 4) satisfying intersection (combined cycle) rules. These
infinite classes of Euler graphs serve as candidates for gracefulness. Infinite
families of graceful graphs are presented in Case-0.
