On Bridge Graphs with Local Antimagic Chromatic Number 3
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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2024-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/1/16 |
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| author | Wai-Chee Shiu Gee-Choon Lau Ruixue Zhang |
| author_facet | Wai-Chee Shiu Gee-Choon Lau Ruixue Zhang |
| author_sort | Wai-Chee Shiu |
| 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. A bijection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mo>→</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mo>|</mo><mi>E</mi><mo>|</mo><mo>}</mo></mrow></semantics></math></inline-formula> is called a local antimagic labeling if, for any two adjacent vertices <i>x</i> and <i>y</i>, <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>x</mi><mo>)</mo></mrow><mo>≠</mo><msup><mi>f</mi><mo>+</mo></msup><mrow><mo>(</mo><mi>y</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><msup><mi>f</mi><mo>+</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>x</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>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the set of edges incident to <i>x</i>. Thus, a local antimagic labeling induces a proper vertex coloring of <i>G</i>, where the vertex <i>x</i> is assigned the color <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>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The local antimagic chromatic number <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></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 labelings of <i>G</i>. In this paper, we present some families of bridge graphs with <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></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula> and give several ways to construct bridge graphs with <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></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-5d3676c61fb849159d88a32aab755fca |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-5d3676c61fb849159d88a32aab755fca2025-01-10T13:17:58ZengMDPI AGMathematics2227-73902024-12-011311610.3390/math13010016On Bridge Graphs with Local Antimagic Chromatic Number 3Wai-Chee Shiu0Gee-Choon Lau1Ruixue Zhang2Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong KongCollege of Computing, Informatics and Mathematics, Universiti Teknologi MARA, Johor Branch, Segamat Campus, Johor 85000, MalaysiaSchool of Mathematics and Statistics, Qingdao University, Qingdao 266071, 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. A bijection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mo>→</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mo>|</mo><mi>E</mi><mo>|</mo><mo>}</mo></mrow></semantics></math></inline-formula> is called a local antimagic labeling if, for any two adjacent vertices <i>x</i> and <i>y</i>, <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>x</mi><mo>)</mo></mrow><mo>≠</mo><msup><mi>f</mi><mo>+</mo></msup><mrow><mo>(</mo><mi>y</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><msup><mi>f</mi><mo>+</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>x</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>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the set of edges incident to <i>x</i>. Thus, a local antimagic labeling induces a proper vertex coloring of <i>G</i>, where the vertex <i>x</i> is assigned the color <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>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The local antimagic chromatic number <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></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 labelings of <i>G</i>. In this paper, we present some families of bridge graphs with <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></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula> and give several ways to construct bridge graphs with <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></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/13/1/16local antimagic labelinglocal antimagic chromatic number<i>s</i>-bridge graphs |
| spellingShingle | Wai-Chee Shiu Gee-Choon Lau Ruixue Zhang On Bridge Graphs with Local Antimagic Chromatic Number 3 Mathematics local antimagic labeling local antimagic chromatic number <i>s</i>-bridge graphs |
| title | On Bridge Graphs with Local Antimagic Chromatic Number 3 |
| title_full | On Bridge Graphs with Local Antimagic Chromatic Number 3 |
| title_fullStr | On Bridge Graphs with Local Antimagic Chromatic Number 3 |
| title_full_unstemmed | On Bridge Graphs with Local Antimagic Chromatic Number 3 |
| title_short | On Bridge Graphs with Local Antimagic Chromatic Number 3 |
| title_sort | on bridge graphs with local antimagic chromatic number 3 |
| topic | local antimagic labeling local antimagic chromatic number <i>s</i>-bridge graphs |
| url | https://www.mdpi.com/2227-7390/13/1/16 |
| work_keys_str_mv | AT waicheeshiu onbridgegraphswithlocalantimagicchromaticnumber3 AT geechoonlau onbridgegraphswithlocalantimagicchromaticnumber3 AT ruixuezhang onbridgegraphswithlocalantimagicchromaticnumber3 |