Linear Forest <i>mP</i><sub>3</sub> Plus a Longer Path <i>P</i><sub>n</sub> Becoming Antimagic
An edge labeling of a graph <i>G</i> is a bijection <i>f</i> from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/12/2036 |
| _version_ | 1827658711187849216 |
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| author | Jen-Ling Shang Fei-Huang Chang |
| author_facet | Jen-Ling Shang Fei-Huang Chang |
| author_sort | Jen-Ling Shang |
| collection | DOAJ |
| description | An edge labeling of a graph <i>G</i> is a bijection <i>f</i> from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> to a set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>|</mo></mrow></semantics></math></inline-formula> integers. For a vertex <i>u</i> in <i>G</i>, the induced vertex sum of <i>u</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>+</mo></msup><mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, is 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 stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. Graph <i>G</i> is said to be antimagic if it has an edge labeling <i>g</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mo>|</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>|</mo><mo stretchy="false">}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>g</mi><mo>+</mo></msup><mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mo>≠</mo><msup><mi>g</mi><mo>+</mo></msup><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> for any pair <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>V</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>≠</mo><mi>v</mi></mrow></semantics></math></inline-formula>. A linear forest is a union of disjoint paths of orders greater than one. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><msub><mi>P</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> denote a linear forest consisting of <i>m</i> disjoint copies of path <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>k</mi></msub></semantics></math></inline-formula>. It is known that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><msub><mi>P</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> is antimagic if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In this study, we add a disjoint path <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>n</mi></msub></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></semantics></math></inline-formula>) to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><msub><mi>P</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> and develop a necessary condition and a sufficient condition whereby the new linear forest <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><msub><mi>P</mi><mn>3</mn></msub><mo>⋃</mo><msub><mi>P</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> may be antimagic. |
| first_indexed | 2024-03-09T23:08:28Z |
| format | Article |
| id | doaj.art-0fd871d15e7645bca4b32734a66b21c3 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T23:08:28Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-0fd871d15e7645bca4b32734a66b21c32023-11-23T17:48:41ZengMDPI AGMathematics2227-73902022-06-011012203610.3390/math10122036Linear Forest <i>mP</i><sub>3</sub> Plus a Longer Path <i>P</i><sub>n</sub> Becoming AntimagicJen-Ling Shang0Fei-Huang Chang1Department of Marketing, Kainan University, Luzhu Dist., Taoyuan City 33857, TaiwanAcademy of Preparatory Programs for Overseas Chinese Students, National Taiwan Normal University, Linkou Dist., New Taipei City 24449, TaiwanAn edge labeling of a graph <i>G</i> is a bijection <i>f</i> from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> to a set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>|</mo></mrow></semantics></math></inline-formula> integers. For a vertex <i>u</i> in <i>G</i>, the induced vertex sum of <i>u</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>+</mo></msup><mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, is 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 stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msub><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. Graph <i>G</i> is said to be antimagic if it has an edge labeling <i>g</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mo>|</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>|</mo><mo stretchy="false">}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>g</mi><mo>+</mo></msup><mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mo>≠</mo><msup><mi>g</mi><mo>+</mo></msup><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> for any pair <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>V</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>≠</mo><mi>v</mi></mrow></semantics></math></inline-formula>. A linear forest is a union of disjoint paths of orders greater than one. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><msub><mi>P</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> denote a linear forest consisting of <i>m</i> disjoint copies of path <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>k</mi></msub></semantics></math></inline-formula>. It is known that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><msub><mi>P</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> is antimagic if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In this study, we add a disjoint path <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>n</mi></msub></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></semantics></math></inline-formula>) to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><msub><mi>P</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> and develop a necessary condition and a sufficient condition whereby the new linear forest <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><msub><mi>P</mi><mn>3</mn></msub><mo>⋃</mo><msub><mi>P</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> may be antimagic.https://www.mdpi.com/2227-7390/10/12/2036edge labelingantimagic labelingantimagic graphdisconnected antimagic graphlinear forest |
| spellingShingle | Jen-Ling Shang Fei-Huang Chang Linear Forest <i>mP</i><sub>3</sub> Plus a Longer Path <i>P</i><sub>n</sub> Becoming Antimagic Mathematics edge labeling antimagic labeling antimagic graph disconnected antimagic graph linear forest |
| title | Linear Forest <i>mP</i><sub>3</sub> Plus a Longer Path <i>P</i><sub>n</sub> Becoming Antimagic |
| title_full | Linear Forest <i>mP</i><sub>3</sub> Plus a Longer Path <i>P</i><sub>n</sub> Becoming Antimagic |
| title_fullStr | Linear Forest <i>mP</i><sub>3</sub> Plus a Longer Path <i>P</i><sub>n</sub> Becoming Antimagic |
| title_full_unstemmed | Linear Forest <i>mP</i><sub>3</sub> Plus a Longer Path <i>P</i><sub>n</sub> Becoming Antimagic |
| title_short | Linear Forest <i>mP</i><sub>3</sub> Plus a Longer Path <i>P</i><sub>n</sub> Becoming Antimagic |
| title_sort | linear forest i mp i sub 3 sub plus a longer path i p i sub n sub becoming antimagic |
| topic | edge labeling antimagic labeling antimagic graph disconnected antimagic graph linear forest |
| url | https://www.mdpi.com/2227-7390/10/12/2036 |
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