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<p>A <i>k</i>-<i>total coloring</i> of a graph <i>G</i> is an assignment of <i>k</i> colors to the vertices and edges of <i>G</i> such that adjacent or incident elements have different colors. The <i>total chromatic number</i> of <i>G</i>, denoted by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44425_2025_16_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi ''(G)\)</EquationSource> </InlineEquation>, is the smallest natural <i>k</i> for which <i>G</i> has a <i>k</i>-total coloring. In 2015, Brinkmann, Preissmann, and Sasaki conjectured that there is no cubic graph with girth at least 5 and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44425_2025_16_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi ''(G) = 5\)</EquationSource> </InlineEquation>. As positive evidence, we prove that several fullerene graphs have&#xa0;<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44425_2025_16_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi ''(G) = 4\)</EquationSource> </InlineEquation>.</p>

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The Total Chromatic Number of Fullerene Molecular Graphs

  • 期刊论文

  • 发表日期：

    2025年11月3日

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  • Mariana Martins,
  • Mauro Nigro,
  • Diana Sasaki

摘要

A k-total coloring of a graph G is an assignment of k colors to the vertices and edges of G such that adjacent or incident elements have different colors. The total chromatic number of G, denoted by \(\chi ''(G)\) , is the smallest natural k for which G has a k-total coloring. In 2015, Brinkmann, Preissmann, and Sasaki conjectured that there is no cubic graph with girth at least 5 and \(\chi ''(G) = 5\) . As positive evidence, we prove that several fullerene graphs have  \(\chi ''(G) = 4\) .