Anti-magic like labelling of marigold graph

K M P G S C Kapuhennayake Kapuhennayake,A A I Perera

doi:10.4038/jsc.v12i2.36

Abstract

In this paper, we present a new family of graphs called Marigold graphs and introduce a new labelling method similar to the anti-magic labelling. The Marigold graph is generated from any number of copies of fully binary trees which are going through concentric circles. All copies of trees are connected to a middle vertex and the height of the Marigold graph is increasing with n concentric circles. One copy is considered as one petal in the marigold graph. A Marigold graph with copies (petals) and height (number of concentric circles) k is M nk denoted by . The labelling method is defined as follows: A graph with ‘m’ edges and ‘n’ vertices is labelled as an injection from the set of edges to the integers {1, …, x} such that all ‘n’ vertex sums are pairwise distinct, where the vertex sum is the sum of labels of all edges incident with that vertex. In our work, for edge labelling, we consider the petals one by one and denote the rth edge at kth level as ekr , and define a function to label edges of the first petal. Then define the new labelling method for other petals, such that for nth petal, edge labelling is starting with Jn–1 + 1 (where Jn–1 is the summation of all edge values in (n –1)th petal, ∑i=1m (n–1, i) = Jn–1 and continue the labelling as a monotonically increasing sequence. We discuss some illustrative examples that might be used for studying the Anti-magic like labelling of Marigold graphs.

Highlights

Graph labelling is one of the important areas in graph theory
When the new labelling method applied, the vertex values of the Marigold graph get increased when we go through the first petal to the last petal in order, and they are increased when go inside to the graph, through the concentric circles
That is the vertices in the outer circle have less values in each copy of perfect binary trees

Summary

Introduction

Graph labelling is one of the important areas in graph theory. Graph labelling is used to give an identity to all the vertices and edges of it. Many variations of anti-magic labelling have been studied by referring to their book They speculated that all simple connected graphs except are anti-magic. Adding an extended to the research chain of Anti-magic labelling, in 2010 Hefetz, Mütze, and Schwartz initiated the study of anti-magic labelling of digraphs. In the last sections of their publication, they conclude that all connected digraphs with at least 4 vertices are antimagic They conjectured that every connected undirected graph admits an antimagic orientation. Mallikarjuna Reddy conjectured that the class of trees generated from two copies of full binary trees is anti-magic. (Reddy & Reddy, 2020)

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Conclusion