When do graph covers preserve the clique dynamics of infinite graphs?
Anna M. Limbach, Martin Winter
TL;DR
The paper investigates when clique dynamics, under the operator $k$, are preserved under triangular covers for locally finite graphs. It first demonstrates a counterexample showing that general cover stability fails: a clique-convergent graph can have both a clique-divergent quotient and a clique-divergent cover. It then identifies two local-condition families that guarantee cover stability: (i) local girth at least $7$ with local minimum degree at least $2$, and (ii) graphs that are locally cyclic with minimum degree at least $6$, and it extends clique-Helly theory to infinite graphs, including constructions of infinite clique-Helly graphs with arbitrary period lengths. Finally, it proves the local-girth criterion, extending finite-case methods to infinite graphs by leveraging clique-Helly properties and structural constraints, while noting an open question at the $7\to6$ boundary and the nuanced behavior of clique dynamics under covers.
Abstract
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: local girth $\geq 7$ and local minimum degree $\geq 2$; being locally cyclic and of minimum degree $\geq 6$.