https://doi.org/10.1016/j.heliyon.2023.e17761

Local total anti-magic chromatic number of graphs

<jats:p>


UDC 519.17



Let <mml:math>
<mml:mrow>
	<mml:mi>G</mml:mi>
	<mml:mo>=</mml:mo>
	<mml:mrow>
		<mml:mo form="prefix">(</mml:mo>
		<mml:mi>V</mml:mi>
		<mml:mo>,</mml:mo>
		<mml:mi>E</mml:mi>
		<mml:mo form="postfix">)</mml:mo>
	</mml:mrow>
</mml:mrow>
</mml:math> be a graph on <mml:math>
<mml:mrow>
	<mml:mi>p</mml:mi>
</mml:mrow>
</mml:math> vertices with no isolated vertices. A bijection <mml:math>
<mml:mrow>
	<mml:mi>f</mml:mi>
</mml:mrow>
</mml:math> from <mml:math>
<mml:mrow>
	<mml:mi>V</mml:mi>
</mml:mrow>
</mml:math> to <mml:math>
<mml:mrow>
	<mml:mo form="prefix">{</mml:mo>
	<mml:mn>1,2</mml:mn>
	<mml:mn>,3</mml:mn>
	<mml:mo>,</mml:mo>
	<mml:mo>…</mml:mo>
	<mml:mo>,</mml:mo>
	<mml:mi>p</mml:mi>
	<mml:mo form="postfix">}</mml:mo>
</mml:mrow>
</mml:math> is called a local distance antimagic labeling if, for any two adjacent vertices <mml:math>
<mml:mrow>
	<mml:mi>u</mml:mi>
</mml:mrow>
</mml:math> and <mml:math>
<mml:mrow>
	<mml:mi>v</mml:mi>
	<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math> we receive distinct weights (colors), where a vertex <mml:math>
<mml:mrow>
	<mml:mi>x</mml:mi>
</mml:mrow>
</mml:math>  has the weight  <mml:math>
<mml:mrow>
	<mml:mi>w</mml:mi>
	<mml:mrow>
		<mml:mo form="prefix">(</mml:mo>
		<mml:mi>x</mml:mi>
		<mml:mo form="postfix">)</mml:mo>
	</mml:mrow>
	<mml:mo>=</mml:mo>
	<mml:mstyle displaystyle="true">
		<mml:msub>
			<mml:mo>∑</mml:mo>
			<mml:mrow>
				<mml:mi>v</mml:mi>
				<mml:mi>ϵ</mml:mi>
				<mml:mi>N</mml:mi>
				<mml:mrow>
					<mml:mo form="prefix">(</mml:mo>
					<mml:mi>x</mml:mi>
					<mml:mo form="postfix">)</mml:mo>
				</mml:mrow>
			</mml:mrow>
		</mml:msub>
	</mml:mstyle>
	<mml:mi>f</mml:mi>
	<mml:mrow>
		<mml:mo form="prefix">(</mml:mo>
		<mml:mi>v</mml:mi>
		<mml:mo form="postfix">)</mml:mo>
	</mml:mrow>
	<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math> The local distance antimagic chromatic number <mml:math>
<mml:mrow>
	<mml:msub>
		<mml:mi>χ</mml:mi>
		<mml:mrow>
			<mml:mi>l</mml:mi>
			<mml:mo>ⅆ</mml:mo>
			<mml:mi>a</mml:mi>
		</mml:mrow>
	</mml:msub>
	<mml:mrow>
		<mml:mo form="prefix">(</mml:mo>
		<mml:mi>G</mml:mi>
		<mml:mo form="postfix">)</mml:mo>
	</mml:mrow>
</mml:mrow>
</mml:math> is defined as the least number of colors used in any local distance antimagic labeling of <mml:math>
<mml:mrow>
	<mml:mi>G</mml:mi>
	<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math> We determine the local distance antimagic chromatic number for the disjoint union of <mml:math>
<mml:mrow>
	<mml:mi>t</mml:mi>
</mml:mrow>
</mml:math> copies of stars and double stars.</jats:p>

Local distance antimagic chromatic number for the union of star and double star graphs

Let G = (V,E) be a graph of order p and size q having no isolated vertices. A bijection ƒ : E → {1, 2, 3, ..., q} is called a local antimagic labeling if for all uv ∈ E we have w(u) ≠ w(v), the weight w(u) = ∑e∈E(u) f(e) where E(u) is the set of edges incident to u. A graph G is local antimagic if G has a local antimagic labeling. The local antimagic chromatic number χla(G) is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we determine the local antimagic chromatic number for some wheel related graphs.

/pdf/local-vertex-antimagic-chromatic-number-of-some-wheel-25lqpfvx.pdf

Local vertex antimagic chromatic number of some wheel related graphs

Let G = (V, E) be a graph of order p and size q without isolated vertices. A bijection f: E → {1, 2, … , q} is called a local antimagic labeling if w(u) ≠ w(v) for all uv ∈ E, where the vertex weight w(u) = Ʃe∈E(u) f(e) and E(u) is the set of edges incident to the vertex u ∈ V. The local antimagic chromatic number χla(G) is defined to be the minimum number of colors(vertex weights) taken over all colorings of G induced by local antimagic labelings of G. In this paper, we study the local chromatic number for generalized friendship graph of complete graphs and cycles. 

Local antimagic vertex coloring for generalized friendship graphs

In this paper we introduce the family of all semi–ℌ –regular in a GTS (X, λ) with a hereditary class ℌ and investigate some of its properties. Also we introduce ℌAB –sets using semi–ℌ –regular set and study some of their properties. 

R. Shankar

Papers

Local vertex antimagic chromatic number of some wheel related graphs

Local antimagic vertex coloring for generalized friendship graphs

Semi–ℌ –regular sets and ℌAB –sets in generalized topological space