Local Antimagic Chromatic Number for Copies of Graphs

An edge labeling of a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> using every label from the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</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>}</mo></mrow></semantics></math></inline-formula> exactly once is a <i>local antimagic labeling</i> if the vertex-weights are distinct for every pair of neighboring vertices, where a vertex-weight is the sum of labels of all edges incident with that vertex. Any <i>local antimagic labeling</i> induces a proper vertex coloring of <i>G</i> where the color of a vertex is its vertex-weight. This naturally leads to the concept of a local antimagic chromatic number. The local antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of <i>G</i> induced by <i>local antimagic labelings</i> of <i>G</i>. In this paper, we estimate the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph.