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<p>Let <i>T</i> be a tree. A vertex of degree one is a <i>leaf</i> of <i>T</i> and a vertex of degree at least three is a <i>branch vertex</i> of <i>T</i>. A graph is said to be <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_568_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(K_{1,4}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> </math></EquationSource> </InlineEquation><i>-free</i> if it does not contain <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_568_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(K_{1,4}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_568_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(K_ {1,4}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>-free graphs. Applying the main results, we also give some improvements of previous results on the spanning tree with few branch vertices for the case of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_568_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(K_{1,4}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>-free graphs.</p>

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Spanning Trees of \(K_{1,4}\)-free Graphs with a Bounded Number of Leaves and Branch Vertices

  • 期刊论文

  • 发表日期：

    2025年4月26日

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  • Pham Hoang Ha

摘要

Let T be a tree. A vertex of degree one is a leaf of T and a vertex of degree at least three is a branch vertex of T. A graph is said to be \(K_{1,4}\) K 1 , 4 -free if it does not contain \(K_{1,4}\) K 1 , 4 as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of \(K_ {1,4}\) K 1 , 4 -free graphs. Applying the main results, we also give some improvements of previous results on the spanning tree with few branch vertices for the case of \(K_{1,4}\) K 1 , 4 -free graphs.