| Summary: | 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.
|
|---|