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
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.
In application/xml+jats
format
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