Eulers Graph World -- More Conjectures On Gracefulness Boundaries-II

TL;DR

This paper explores specific subclasses of Euler graphs with cycle types under mod 4, investigates their properties, and proposes conjectures to better understand the boundaries of graph gracefulness.

Contribution

It introduces new classes of Euler graphs with two cycle types, provides constructions, and formulates conjectures on their gracefulness boundaries.

Findings

Regular bipartite Euler graphs are well known.

Certain cycle type combinations lead to nonexistence of regular Euler graphs.

Some cases produce planar graphs, others may be nonplanar.

Abstract

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.