We propose a Law of Nature? Viz., Pure Regularity Occurs at Naïve 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.