Extending of Edge Even Graceful Labeling of Graphs to Strong r -Edge Even Graceful Labeling release_7yqlffq2qvfjdn5hgfjpawobou

by mohamed Zeen El-Deen, Nora A. Omar

Published in Journal of Mathematics by Hindawi Limited.

2021   Volume 2021, p1-19

Abstract

Edge even graceful labeling of a graph <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>G</mi> </math> </jats:inline-formula> with <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>p</mi> </math> </jats:inline-formula> vertices and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>q</mi> </math> </jats:inline-formula> edges is a bijective <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <mi>f</mi> </math> </jats:inline-formula> from the set of edge <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mi>E</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>G</mi> </mrow> </mfenced> </math> </jats:inline-formula> to the set of positive integers <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <mfenced open="{" close="}" separators="|"> <mrow> <mn>2,4</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>2</mn> <mi>q</mi> </mrow> </mfenced> </math> </jats:inline-formula> such that all the vertex labels <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <msup> <mrow> <mi>f</mi> </mrow> <mi>∗</mi> </msup> <mfenced open="[" close="]" separators="|"> <mrow> <mi>V</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>G</mi> </mrow> </mfenced> </mrow> </mfenced> </math> </jats:inline-formula>, given by <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9"> <msup> <mrow> <mi>f</mi> </mrow> <mi>∗</mi> </msup> <mfenced open="(" close=")" separators="|"> <mrow> <mi>u</mi> </mrow> </mfenced> <mo>=</mo> <mfenced open="(" close=")" separators="|"> <mrow> <mstyle displaystyle="true"> <msub> <mrow> <mo stretchy="false">∑</mo> </mrow> <mrow> <mi>u</mi> <mi>v</mi> <mo>∈</mo> <mi>E</mi> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi>G</mi> </mrow> </mfenced> </mrow> </mrow> </msub> <mrow> <mi>f</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>u</mi> <mi>v</mi> </mrow> </mfenced> </mrow> </mstyle> </mrow> </mfenced> <mi mathvariant="normal">mod</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </mfenced> </math> </jats:inline-formula>, where <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10"> <mi>k</mi> <mo>=</mo> <mi mathvariant="normal">max</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </mfenced> </math> </jats:inline-formula>, are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11"> <mi>r</mi> </math> </jats:inline-formula>-edge even graceful labeling and strong <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M12"> <mi>r</mi> </math> </jats:inline-formula>-edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M13"> <mi>r</mi> </math> </jats:inline-formula>-edge even graceful graph. Furthermore, the minimum number <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M14"> <mi>r</mi> </math> </jats:inline-formula> for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M15"> <mi>r</mi> </math> </jats:inline-formula>-edge even graceful labeling was found. Finally, we proved that the even cycle <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M16"> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mi>n</mi> </math> </jats:inline-formula> has a strong <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M17"> <mn>2</mn> </math> </jats:inline-formula>-edge even graceful labeling when <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M18"> <mi>n</mi> </math> </jats:inline-formula> is even.
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