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<p>We investigate for which classes of (potentially infinite) graphs the clique dynamics is cover stable, i.&#xa0;e., when clique convergence/divergence is preserved under triangular covering maps. We first present an instructive counterexample: a clique convergent graph, which covers a clique divergent graph and which is covered by a clique divergent graph. Based on this, we then focus on local conditions (i.&#xa0;e., conditions on the neighbourhoods of vertices) and show that the following are sufficient to imply cover stability for connected graphs: local girth <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44425_2025_21_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ge 7\)</EquationSource> </InlineEquation> and local minimum degree <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44425_2025_21_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ge 2\)</EquationSource> </InlineEquation>; being locally cyclic and of minimum degree <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44425_2025_21_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ge 6\)</EquationSource> </InlineEquation>.</p>

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When Do Graph Covers Preserve the Clique Dynamics of Infinite Graphs?

  • 期刊论文

  • 发表日期：

    2025年9月30日

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  • Anna Margarethe Limbach,
  • Martin Winter

摘要

We investigate for which classes of (potentially infinite) graphs the clique dynamics is cover stable, i. e., when clique convergence/divergence is preserved under triangular covering maps. We first present an instructive counterexample: a clique convergent graph, which covers a clique divergent graph and which is covered by a clique divergent graph. Based on this, we then focus on local conditions (i. e., conditions on the neighbourhoods of vertices) and show that the following are sufficient to imply cover stability for connected graphs: local girth \(\ge 7\) and local minimum degree \(\ge 2\) ; being locally cyclic and of minimum degree \(\ge 6\) .