<p>An antimagic labeling of a graph <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1828_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(G=(V,E)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of size <i>m</i> is a one-to-one mapping <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1828_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="163" /> </InlineMediaObject> <EquationSource Format="TEX">\(f: E\rightarrow \{1,2,\ldots ,m\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, such that all vertices receive pairwise distinct vertex sums, where a vertex sum of a vertex <i>v</i> in <i>G</i> is the sum of the labels on the edges incident to <i>v</i>. A graph is called antimagic if it admits an antimagic labeling. It was conjectured that every connected graph other than <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1828_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(K_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> is antimagic by Hartsfield et al. in 1990. The conjecture remains unsolved, which seems hard for sparse graphs, such as trees, especially for those trees with many vertices of degree 2. A subdivided caterpillar is a tree obtained from a caterpillar by subdividing each leg the same number of times. This paper shows that every subdivided caterpillar is antimagic.</p>

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Antimagic labeling of subdivided caterpillars

  • Canbin Wu,
  • Kecai Deng,
  • Qinghong Zhao

摘要

An antimagic labeling of a graph \(G=(V,E)\) G = ( V , E ) of size m is a one-to-one mapping \(f: E\rightarrow \{1,2,\ldots ,m\}\) f : E { 1 , 2 , , m } , such that all vertices receive pairwise distinct vertex sums, where a vertex sum of a vertex v in G is the sum of the labels on the edges incident to v. A graph is called antimagic if it admits an antimagic labeling. It was conjectured that every connected graph other than \(K_2\) K 2 is antimagic by Hartsfield et al. in 1990. The conjecture remains unsolved, which seems hard for sparse graphs, such as trees, especially for those trees with many vertices of degree 2. A subdivided caterpillar is a tree obtained from a caterpillar by subdividing each leg the same number of times. This paper shows that every subdivided caterpillar is antimagic.