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

TL;DR

This paper explores intrinsic properties and conjectures related to Euler graphs, focusing on their cycle structures and gracefulness boundaries, aiming to guide future research in graph gracefulness characterization.

Contribution

It introduces new properties of Euler graphs, proposes gracefulness boundaries for subclasses, and updates existing conjectures to aid in understanding graph gracefulness.

Findings

Euler graphs with higher regularity than degree two are impossible within certain cycle classes.

Some subclasses of Euler graphs are planar for specific cycle types.

Proposed gracefulness boundaries may guide future graph characterization efforts.

Abstract

Euler graphs are characterized by the simple criterion that degree of each node is even. By restricting on the cycle types yet additional intrinsic properties of Euler graphs are unveiled. For example, regularity higher than degree two is impossible within the class {\epsilon}i of Euler graphs with one type of cycles Cn, n=i(mod 4), i=0,1,2,3. Further, graphs in {\epsilon}i are planar for i=1,2,3. In the light of new properties of Euler graphs more gracefulness boundaries are conjectured for subclasses of Euler graphs and where relevant extended for general class of graphs. In absence of general analytical results much of the published papers resort to proving an infinite class of graphs graceful or nongraceful. The purpose of this paper is not to give families of graphs graceful or not. Instead, based on the available information expected gracefulness boundaries are proposed which may guide where to look for graceful graphs or lead to characterizations. While the (Ringel,Kotzig,Rosa) Tree Conjecture continues to remain unsettled, the work reported here serves an update on the conjectures made in Rao Hebbare (1975,1981) and more conjectures subsequently made in Rao (1999,2000) based on embedding theorems and graceful algorithms for constructing graceful graphs from a graceful graph. It is hoped that these conjectures lead to analytical techniques for establishing gracefulness property. Further probe into Euler graphs with only two types of cycles and other combinations of cycles continues.