Further Results on Bijective Product <i>k</i>-Cordial Labeling
A bijective product <i>k</i>-cordial labeling <i>f</i> of a graph <i>G</i> with vertex set <i>V</i> and edge set <i>E</i> is a bijection from <i>V</i> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/Mat...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-07-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/15/2451 |
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| Summary: | A bijective product <i>k</i>-cordial labeling <i>f</i> of a graph <i>G</i> with vertex set <i>V</i> and edge set <i>E</i> is a bijection from <i>V</i> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mo stretchy="false">|</mo><mi>V</mi><mo stretchy="false">|</mo><mo>}</mo></mrow></semantics></math></inline-formula> such that the induced edge labeling <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>×</mo></msup><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><msub><mi mathvariant="double-struck">Z</mi><mi>k</mi></msub><mo>=</mo><mrow><mo>{</mo><mi>i</mi><mspace width="0.277778em"></mspace><mo stretchy="false">|</mo><mspace width="0.277778em"></mspace><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>×</mo></msup><mrow><mo>(</mo><mi>u</mi><mi>v</mi><mo>)</mo></mrow><mo>≡</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mspace width="4.44443pt"></mspace><mrow><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.277778em"></mspace><mi>k</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> for every edge <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi></mrow></semantics></math></inline-formula> satisfies the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">|</mo></mrow><msubsup><mi>e</mi><mi>f</mi><mo>×</mo></msubsup><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>−</mo><msubsup><mi>e</mi><mi>f</mi><mo>×</mo></msubsup><mrow><mrow><mo>(</mo><mi>j</mi><mo>)</mo></mrow><mo stretchy="false">|</mo><mo>≤</mo><mn>1</mn></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><msub><mi mathvariant="double-struck">Z</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>e</mi><mi>f</mi><mo>×</mo></msubsup><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the number of edges labeled with <i>i</i> under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>f</mi><mo>×</mo></msup></semantics></math></inline-formula>. A graph which admits a bijective product <i>k</i>-cordial labeling is called a bijective product <i>k</i>-cordial graph. In this paper, we study bijective product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>-cordiality for paths and cycles, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> is an odd prime. We determine bijective product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>-cordiality for paths and cycles for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>≤</mo><mi>π</mi><mo>≤</mo><mn>13</mn></mrow></semantics></math></inline-formula>. Also, we establish the bijective product <i>k</i>-cordial labeling of stars. Further, we find the bijective product 4-cordial labeling of bistars and the splitting graphs of stars and bistars. |
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| ISSN: | 2227-7390 |