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

TL;DR

This paper explores Euler graphs with three cycle types under mod 4, providing constructions, conjectures on degree nodes and regularity, and analyzing gracefulness boundaries for these complex graph classes.

Contribution

It introduces new classes of Euler graphs with three cycle types, proves existence of degree two nodes in some cases, and conjectures nonexistence of regular graphs of order p>5 with these cycle types.

Findings

Constructed Euler graphs with three cycle types

Proved existence of degree two nodes in some cases

Conjectured nonexistence of regular Euler graphs of order p>5 with three cycle types

Abstract

Euler graphs with only one (two) type(s) of cycles under (mod 4) operation were studied in Part-I(II). Here we consider the class of Euler graphs with only three types of cycles under (mod 4). This gives rise to four cases viz., graphs having cycle types (0,1,2), (0,1,3), (0,2,3), (1,2,3). Some constructions of Euler graphs of each class are given. We prove the existence of degree two node in part cases and conjecture the existence of degree two node in every graph of order p>5. As a special case, we conjecture that regularity is nonexistent in all four cases for graphs of order p>5. That is, regular Euler graphs of order p>5 with exactly three types of cycles donot exist. In other words, a regular Euler graph of order p>5 with three types of cycles has a cycle of fourth type. An example of a nonplanar Euler graph in the first three cases is given. Graphs of this type satisfying Rosa-Golomb criterion are nongraceful. In other cases necessary conditions are given and the graphs are conjectured graceful leading to better understanding of gracefulness boundaries.