The Edge Odd Graceful Labeling of Water Wheel Graphs

The Edge Odd Graceful Labeling of Water Wheel Graphs

A graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</...

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Bibliographic Details
Main Authors:	Mohammed Aljohani, Salama Nagy Daoud
Format:	Article
Language:	English
Published:	MDPI AG 2024-12-01
Series:	Axioms
Subjects:	graceful labeling edge graceful labeling edge odd graceful labeling water wheel graph
Online Access:	https://www.mdpi.com/2075-1680/14/1/5
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Summary:	A graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is edge odd graceful if it possesses edge odd graceful labeling. This labeling is defined as a bijection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>:</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>, from which an injective transformation is derived, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>g</mi><mo></mo></msup><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, from the rule that the image of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>g</mi><mo></mo></msup></semantics></math></inline-formula> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mi>g</mi><mrow><mo>(</mo><mi>u</mi><mi>v</mi><mo>)</mo></mrow><mspace width="0.277778em"></mspace><mi>mod</mi><mspace width="0.277778em"></mspace><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The main objective of this manuscript is to introduce new classes of planar graphs, namely water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><msub><mi>W</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>; triangulated water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><msub><mi>W</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>; closed water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>W</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>; and closed triangulated water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>T</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>. Furthermore, we specify conditions for these graphs to allow for edge odd graceful labelings.
ISSN:	2075-1680