On Bridge Graphs with Local Antimagic Chromatic Number 3

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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Bibliographic Details
Main Authors:	Wai-Chee Shiu, Gee-Choon Lau, Ruixue Zhang
Format:	Article
Language:	English
Published:	MDPI AG 2024-12-01
Series:	Mathematics
Subjects:	local antimagic labeling local antimagic chromatic number <i>s</i>-bridge graphs
Online Access:	https://www.mdpi.com/2227-7390/13/1/16
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Summary:	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>.
ISSN:	2227-7390