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

TL;DR

This work characterizes clique convergence for locally cyclic graphs with minimum degree $δ≥6$ by linking it to the global topological structure of the universal triangular cover. For triangularly simply connected cases, clique divergence occurs exactly when arbitrarily large triangular-shaped subgraphs $oldsymbol{ riangle_m}$ appear, shown via a finite-but-unbounded invariant based on degree-26 vertices. The general case is reduced to the universal cover using group actions and quotient graphs, establishing a cohesive criterion: a connected locally cyclic graph $G$ with $δ≥6$ is clique divergent if and only if its universal triangular cover contains arbitrarily large $oldsymbol{ riangle_m}$. This connects topological obstructions to the long-term behavior of clique dynamics, and confirms divergence for the hexagonal lattice and its quotients. The results unify local structure with global geometry and suggest avenues for extending the theory to lower degrees and higher-dimensional triangulations.

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 $δ\ge 6$, 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".

Figures (14)

• Figure 1: The tri-an-gu-lar-shaped graphs $\Delta_m$ for $m\in \{0,\ldots,4\}$.
• Figure 2: The tri-an-gu-lar-shaped graph $\Delta_4$ and its boundary $\partial\Delta_4$.
• Figure 3: The 26 possible ways in which a tri-an-gu-lar-shaped graph $S\in V(G_n)$ of side length $m\ge 6$ can be $G_n$-adjacent to another tri-an-gu-lar-shaped graph $T\in V(G_n)$ of side length $m+s$, where $s\in\{-6,-4,-2,0,+2,+4,+6\}$. Two configurations may differ merely by a symmetry (one of the six "reflections" and "rotations" of a tri-an-gu-lar-shaped graph), and we always show only a single configuration with the multiplication factor next to it indicating the number of equivalent configuration related by symmetry. Note that for the types $\pm2$, $\pm4$ and $\pm6$, the configurations must be accounted for twice in the $G_n$-degree of $S$: once with $S$ being the larger graph (in grey), and once with $S$ being the smaller graph (in black). Then $26=6+2\cdot(3+3+3+1)$.
• Figure 4: For $m\in\{2,4\}$, there also exist the following "twisted adjacencies".
• Figure 5: The eight possible neighbours of a tri-an-gu-lar-shaped graph of side length $m=0$ of type $+4$ and $+6$. See the caption of \ref{['fig:large_enough_cases']} for an explanation of the multiplicities.

Theorems & Definitions (44)

• Theorem A: Characterisation theorem for triangularly simply connected graphs
• Theorem B: General characterisation theorem
• Definition 2.1: BAUMEISTER2022112873
• Theorem 2.2: BAUMEISTER2022112873
• Example 2.3
• Remark 3.1
• Lemma 3.2
• proof : Proof of \ref{['res:m_ge_6_degrees_26_iff']}
• Lemma 3.3
• proof