Eulers Graph World - Purity, Regularity and Evenness -Law of Nature? - Constructions and Examples

TL;DR

This paper explores the properties of Euler graphs with respect to cycle types, regularity, and evenness, proposing a 'Law of Nature' that relates pure regularity to naive levels and evenness, with new constructions and conjectures.

Contribution

It introduces new conjectures about the existence of regular Euler graphs with specific cycle types and provides constructions of Euler graphs satisfying certain intersection rules.

Findings

K5 is the only known regular Euler graph with three cycle types

Regular Euler graphs with only two cycle types exist in specific bipartite cases

Infinite classes of Euler graphs are constructed as candidates for graceful graphs

Abstract

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.