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arXiv · 2305.00503

Characterising Clique Convergence for Locally Cyclic Graphs of Minimum Degree $\delta\ge 6$

Anna M. Limbach·Martin Winter
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Abstract

The clique graph $kG$ of a graph $G$ has as its vertices the cliques (maximal complete subgraphs) of $G$, two of which are adjacent in $kG$ if they have non-empty intersection in $G$. We say that $G$ is clique convergent if $k^nG\cong k^m G$ for some $n\not= m$, and that $G$ is clique divergent otherwise. We completely characterise the clique convergent graphs in the class of (not necessarily finite) locally cyclic graphs of minimum degree $\delta\ge 6$, showing that for such graphs clique divergence is a global phenomenon, dependent on the existence of large substructures. More precisely, we establish that such a graph is clique divergent if and only if its universal triangular cover contains arbitrarily large members from the family of so-called "triangular-shaped graphs".

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Anna M. Limbach, Martin Winter. 2023-04-30. Characterising Clique Convergence for Locally Cyclic Graphs of Minimum Degree $\delta\ge 6$. https://arxiv.org/abs/2305.00503

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