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<p>In a graph <i>G</i>,&#xa0; a proper vertex coloring is called a dominated coloring if, for every color class, there exists at least one vertex that dominates the class. The dominated chromatic number, denoted by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3207_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi _{dom} (G),\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>χ</mi> <mrow> <mi mathvariant="italic">dom</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> is the minimum number of colors required to dominated <i>G</i>. In this study, we determine the dominated chromatic number for several tree structures, such as <i>m</i>-ary tree, sibling tree, hypertree, slim tree, and generalized fat tree.</p>

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Dominated coloring in certain trees

  • 期刊论文

  • 发表日期：

    2025年4月29日

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  • S. Poonkuzhali,
  • R. Jayagopal

摘要

In a graph G,  a proper vertex coloring is called a dominated coloring if, for every color class, there exists at least one vertex that dominates the class. The dominated chromatic number, denoted by \(\chi _{dom} (G),\) χ dom ( G ) , is the minimum number of colors required to dominated G. In this study, we determine the dominated chromatic number for several tree structures, such as m-ary tree, sibling tree, hypertree, slim tree, and generalized fat tree.