The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs
Let <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>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/11/3/97 |
| _version_ | 1827649892165615616 |
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| author | Xue Yang Hong Bian Haizheng Yu Dandan Liu |
| author_facet | Xue Yang Hong Bian Haizheng Yu Dandan Liu |
| author_sort | Xue Yang |
| collection | DOAJ |
| description | Let <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> be a connected graph with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>V</mi><mo>|</mo><mo>=</mo><mi>n</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>E</mi><mo>|</mo><mo>=</mo><mi>m</mi></mrow></semantics></math></inline-formula>. A bijection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>∪</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>}</mo></mrow></semantics></math></inline-formula> is called local antimagic total labeling if, for any two adjacent vertices <i>u</i> and <i>v</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≠</mo><msub><mi>ω</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><msub><mo>∑</mo><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the set of edges incident to <i>u</i>. Thus, any local antimagic total labeling induces a proper coloring of <i>G</i>, where the vertex <i>x</i> in <i>G</i> is assigned the color <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The local antimagic total chromatic number, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>χ</mi><mrow><mi>l</mi><mi>a</mi><mi>t</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, is the minimum number of colors taken over all colorings induced by local antimagic total labelings of <i>G</i>. In this paper, we present the local antimagic total chromatic numbers of some wheel-related graphs, such as the fan graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>n</mi></msub></semantics></math></inline-formula>, the bowknot graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula>, the Dutch windmill graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>D</mi><mrow><mn>4</mn></mrow><mi>n</mi></msubsup></semantics></math></inline-formula>, the analogous Dutch graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><msubsup><mi>D</mi><mrow><mn>4</mn></mrow><mi>n</mi></msubsup></mrow></semantics></math></inline-formula> and the flower graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">F</mi><mi>n</mi></msub></semantics></math></inline-formula>. |
| first_indexed | 2024-03-09T20:06:51Z |
| format | Article |
| id | doaj.art-88d111ed77304f8eaa9e1bb4cb1b6c5f |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T20:06:51Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-88d111ed77304f8eaa9e1bb4cb1b6c5f2023-11-24T00:28:46ZengMDPI AGAxioms2075-16802022-02-011139710.3390/axioms11030097The Local Antimagic Total Chromatic Number of Some Wheel-Related GraphsXue Yang0Hong Bian1Haizheng Yu2Dandan Liu3School of Mathematical Sciences, Xinjiang Normal University, Urumqi 830017, ChinaSchool of Mathematical Sciences, Xinjiang Normal University, Urumqi 830017, ChinaCollege of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, ChinaSchool of Mathematical Sciences, Xinjiang Normal University, Urumqi 830017, ChinaLet <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> be a connected graph with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>V</mi><mo>|</mo><mo>=</mo><mi>n</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>E</mi><mo>|</mo><mo>=</mo><mi>m</mi></mrow></semantics></math></inline-formula>. A bijection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>∪</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>}</mo></mrow></semantics></math></inline-formula> is called local antimagic total labeling if, for any two adjacent vertices <i>u</i> and <i>v</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≠</mo><msub><mi>ω</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><msub><mo>∑</mo><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the set of edges incident to <i>u</i>. Thus, any local antimagic total labeling induces a proper coloring of <i>G</i>, where the vertex <i>x</i> in <i>G</i> is assigned the color <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The local antimagic total chromatic number, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>χ</mi><mrow><mi>l</mi><mi>a</mi><mi>t</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, is the minimum number of colors taken over all colorings induced by local antimagic total labelings of <i>G</i>. In this paper, we present the local antimagic total chromatic numbers of some wheel-related graphs, such as the fan graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>n</mi></msub></semantics></math></inline-formula>, the bowknot graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula>, the Dutch windmill graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>D</mi><mrow><mn>4</mn></mrow><mi>n</mi></msubsup></semantics></math></inline-formula>, the analogous Dutch graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><msubsup><mi>D</mi><mrow><mn>4</mn></mrow><mi>n</mi></msubsup></mrow></semantics></math></inline-formula> and the flower graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">F</mi><mi>n</mi></msub></semantics></math></inline-formula>.https://www.mdpi.com/2075-1680/11/3/97local antimagic total labelinglocal antimagic total chromatic numberwheel-related graphs |
| spellingShingle | Xue Yang Hong Bian Haizheng Yu Dandan Liu The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs Axioms local antimagic total labeling local antimagic total chromatic number wheel-related graphs |
| title | The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs |
| title_full | The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs |
| title_fullStr | The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs |
| title_full_unstemmed | The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs |
| title_short | The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs |
| title_sort | local antimagic total chromatic number of some wheel related graphs |
| topic | local antimagic total labeling local antimagic total chromatic number wheel-related graphs |
| url | https://www.mdpi.com/2075-1680/11/3/97 |
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