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

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".


Introduction
Given a (not necessarily finite) simple graph G, a clique Q ⊆ G is an inclusion maximal complete subgraph.The clique graph kG has as its vertices the cliques of G, two of which are adjacent in kG if they have a non-empty intersection in G.The operator k is known as the clique graph operator and the behaviour of the sequence G, kG, k 2 G, . . . is the clique dynamics of G.The graph is clique convergent if the clique dynamics cycles eventually and it is clique divergent otherwise.It is an ongoing endeavour to understand which graph properties lead to convergence and divergence respectively, however, since clique convergence is known to be undecidable in general [2], this investigation often restricts to certain graph classes, such as graphs of low degree [17], circular arc graphs [13], or locally H graphs (e. g. locally cyclic graphs [6] or shoal graphs [4]).
The focus of the present article is on locally cyclic graphs, that is, graphs for which the neighbourhood of each vertex induces a cycle.Such graphs can be interpreted as triangulations of surfaces (always to be understood as "without boundary"), and it was recognized early that the study of their clique dynamics can be informed by topological considerations.So it is known that each closed surface (i.e., compact and without boundary) has a clique divergent triangulation [9], but that convergent triangulations exist on all closed surfaces of negative Euler characteristic [7].It has furthermore been conjectured that there are no convergent triangulations on closed surfaces of non-negative Euler characteristic (for a precise statement one requires minimum degree δ ≥ 4; see Conjecture 5.1).For example, the 4-regular [3] and 5-regular [14] triangulations of the sphere (i.e., the octahedral and icosahedral graph) are clique divergent; as is any 6-regular triangulation of the torus or Klein bottle [5,6].On the other hand, a triangulation of minimum degree δ ≥ 7 (necessarily of a closed surface of higher genus) is clique convergent [8].Triangulations that mix degrees above and below six are still badly understood.
Baumeister & Limbach [1] broadened these investigations to triangulations of noncompact surfaces, that is, to infinite locally cyclic graphs.They gave an explicit description of k n G in terms of so-called triangular-shaped subgraphs of G (see Figure 1), where G is a triangulation of minimum degree δ ≥ 6 of a (not necessarily compact) simply connected surface (see Section 2.3 for details).
... The goal of this article is to bring the investigation of [1] to a satisfying conclusion: we apply their explicit construction of k n G to completely characterise the clique convergent triangulations in the class of (not necessarily finite) locally cyclic graphs of minimum degree δ ≥ 6.We thereby answer the open questions from Section 9 of [1].
Our first main result concerns locally cyclic graphs that are triangularly simply connected, that is, they correspond to triangulations of simply connected surfaces (see Section 4.2 for a rigorous definition).We identify the clique divergence of these graphs as a consequence of the existence of arbitrarily large triangular-shaped subgraphs.
Theorem A (Characterisation theorem for triangularly simply connected graphs).A triangularly simply connected locally cyclic graph of minimum degree δ ≥ 6 is clique divergent if and only if it contains arbitrarily large triangular-shaped subgraphs.
The difficulty in proving Theorem A lies in establishing divergence for a sequence of infinite graphs.Divergence is usually shown by observing the divergence of some numerical graph parameter, such as the vertex count or graph diameter.As our graphs are potentially infinite, this fails since the straightforward quantities might be infinite to begin with.The quest then lies mainly in identifying an often more contrived graph invariant which is still finite yet unbounded.
As a consequence of Theorem A we find that the 6-regular triangulation of the Euclidean plane (aka.the hexagonal lattice) is clique divergent.
By applying Theorem A to the universal triangular cover (see Section 4.2), we obtain the following more general result.
Theorem B (General characterisation theorem).A (not necessarily finite) connected locally cyclic graph of minimum degree δ ≥ 6 is clique divergent if and only if its universal triangular cover contains arbitrarily large triangular-shaped subgraphs.
The "only if" direction of Theorem B was supposedly proven in [1], but the proof contains a gap, which we close in Section 4.
As a consequence of Theorem B, a triangulation of of minimum degree δ ≥ 6 of a closed surface is clique divergent if and only if it is 6-regular (cf.[1,Lemma 8.10]).
We mention two further recent results on clique dynamics that are in a similar spirit.In 2017, Larrión, Pizaña, and Villarroel-Flores [12] showed that the clique operator preserves (finite) triangular graph bundles, which are a generalisation of finite triangular covering maps.Also, just recently in 2022, Villarroel-Flores [17] showed that among the (finite) connected graphs with maximum degree at most four, the octahedral graph is the only one that is clique divergent.

Structure of the Paper
In Section 2, we recall the fundamental concepts and notations used throughout the paper.In particular, in Section 2.3 we recall the geometric clique graph G n and the relevant statements of [1] that established the explicit description of k n G in terms of G n .
In Section 3, we prove Theorem A. To show that a sequence of infinite graphs is divergent, we identify a finite yet unbounded graph invariant D(H) (see (3.1)) based on the distribution of vertices of degree 26.
In Section 4, we prove Theorem B. We extend the divergence results of Section 3 to graphs that are not necessarily triangularly simply connected by exploiting that covering relations interact well with the clique operator and the geometric clique graph.
Section 5 summarizes the results and lists related open questions.We also include an appendix which recalls helpful background theory for Section 4. Appendix A gives a proof that triangular simple connectivity is preserved under the clique operator while Appendix B focuses on the existence and uniqueness of a triangularly simply connected triangular cover for any connected graph.

Basic Notation
All graphs in this article are simple, non-empty and potentially infinite.If not stated otherwise, they are connected and locally finite.For a graph G we write V (G) and E(G) to denote its vertex set and edge set, respectively.The adjacency relation is denoted by ∼.We define the closed and the open neighbourhood of a set U ⊆ V (G) of vertices as We write G-degree, G-neighbourhood, or G-distance to emphasize the graph with respect to which these quantities are computed.Finally, we use ∼ = to denote isomorphy between graphs.
We write N := {1, 2, 3, . ..} and N 0 := N ∪ {0} for the sets of natural numbers without and with zero.We write kN and kN 0 to denote multiples of k.

Cliques, Clique Graphs, and Clique Dynamics
A clique in G is an inclusion maximal complete subgraph.The clique graph kG has vertex set V (kG) := {cliques of G}, and distinct cliques Q, Q ∈ V (kG) are adjacent in kG if they have vertices in common.We consider k as an operator, the clique graph operator, mapping a graph to its clique graph.By k n , we denote its n-th iterate.
A sequence G 0 , G 1 , G 2 , . . . of graphs is said to be convergent if it is eventually periodic, that is, if for some r ∈ N and all sufficiently large n ∈ N we have G n ∼ = G n+r .The sequence is said to be divergent otherwise.A graph G is said to be clique convergent if the sequence k 0 G, k 1 G, k 2 G, . . . is convergent, and is called clique divergent otherwise.

Locally Cyclic Graphs, Triangular-Shaped Subgraphs, and the Geometric Clique Graph
A graph G is locally cyclic if the (open) neighbourhood of each vertex induces a cycle.
In particular, a locally cyclic graph is locally finite.Such graphs can also be interpreted as triangulations of surfaces.We shall however use this geometric perspective only informally, and work with the purely graph theoretic definition given above.A fundamental example of a locally cyclic graph is the hexagonal triangulation of the Euclidean plane.
We use the class of triangular-shaped graphs ∆ m from [1], which are subgraphs of the hexagonal lattice, and the smallest five of which are depicted in Figure 1.The parameter m is called the side length of ∆ m , and the boundary ∂∆ m is the subgraph of ∆ m that consists of the vertices of degree less than six and the edges that lie in only a single triangle (Figure 2).In [1], it was shown that the n-th iterated clique graph k n G of a triangularly simply connected locally cyclic graph G of minimum degree δ ≥ 6 (also called "pika" in [1]) can be explicitly constructed based on triangular-shaped subgraphs of G (see Definition 2.1 and Theorem 2.2 below).Hereby "triangularly simply connected" means "triangulation of a simply connected surface", but a precise definition is postponed until Section 4.2 (or see [1]).For now it suffices to use this terms as a black box, merely to apply Theorem 2.2.Note however that such a graph is in particular connected.
The explicit construction of k n G is captured by the following definition: We refer to the four types of adjacencies listed in Definition 2.1 as adjacencies of type ±0, ±2, ±4 and ±6 respectively.For a triangular-shaped graph S ∈ V (G n ) of side length m, we refer to a neighbour T ∈ N Gn (S) of side length m + s as being of type s ∈ {−6, −4, −2, ±0, +2, +4, +6}.Some visualisations for the various configurations of triangular-shaped graphs that correspond to adjacency in G n can be seen in Figures 3 to 5 in the next section.
The following example demonstrates how Theorem 2.2 can be used to establish clique convergence in non-trivial cases: Example 2.3.A locally cyclic and triangularly simply connected graph G of minimum degree δ ≥ 7 does not contain any triangular-shaped graphs of side length ≥ 3 (because such have vertices of degree six).Hence, Such a graph G is therefore clique convergent.

Proof of Theorem A
Throughout this section, we assume that G is a locally cyclic graph that is triangularly simply connected and has minimum degree δ ≥ 6.We can then apply Theorem 2.2 and investigate the dynamics of the sequence of geometric clique graphs G n in place of k n G.
One direction of Theorem A follows immediately from the definition of the geometric clique graph (Definition 2.1): if all triangular-shaped subgraphs of G are of side length ≤ m ∈ 2N, then G m ∼ = G m+2 , and the sequence cycles.
The remainder of this section is devoted to proving the other direction of Theorem A: if G contains arbitrarily large triangular-shaped subgraphs, then G is clique divergent.For this, we identify a graph invariant that is both finite and unbounded for the sequence G n as n → ∞, as long as G contains arbitrarily large triangular-shaped subgraphs.It turns out that a suitable graph invariant can be built from measuring distances between vertices of certain degrees.Curiously, the degree 26 plays a special role, and the following notation comes in handy: The corresponding graph invariant is the following: The significance of the number 26 stems from the observation that most vertices of G n have G n -degree ≤ 26; and have G n -degree exactly 26 only in very special circumstances that can be expressed as the existence of certain triangular-shaped subgraphs in G.This is proven in Lemma 3.1 and Lemma 3.2.Finitude and divergence of D(G n ) as n → ∞ are proven afterwards in Lemma 3.3 and Lemma 3.4.
In the following, we generally consider G n only for even n ∈ 2N, as this cuts down on the cases we need to investigate, and is still sufficient to show that D(G n ) is unbounded.Note that each S ∈ V (G n ) is then of even side length m ∈ {0, 2, 4, 6, . ..}. Lemma 3.1.Let S ∈ V (G n ) be a triangular-shaped graph of side length m ≥ 6.Then deg Gn (S) ≤ 26, with equality if and only if S has a neighbour of type +6.Lemma 3.1 actually holds unchanged for m ≥ 2. Since we do not need these cases to prove Theorem A, and since verifying them requires a distinct case analysis (because of "twisted adjacencies", cf. Figure 4), we do not include them here.
Proof of Lemma 3.1.Figure 3 shows all potential configurations of S and a G n -neighbour of S according to Definition 2.1 (here we need m ≥ 6, as there are exceptional "twisted adjacencies" for smaller m, see Figure 4).In total this amounts to a degree of at most 26.In particular, if just one of the neighbours is missing, say the neighbour of type +6, then S must have a G n -degree of less than 26.
Conversely, one can verify that if S has a neighbour of type +6, say T ∈ N Gn (S), then all other neighbours of types −6, −4, −2, 0, +2, and +4 can be found as subgraphs of T .Therefore, all 26 neighbours are present and the degree is 26.For m = 0 only one direction holds, which is also sufficient for our purpose.
Proof.Clearly, s has no neighbours of type −6, −4 or −2.The G n -neighbours of type 0 are exactly the vertices that are also adjacent to s in G, that is, there are exactly deg G (s) many.The potential neighbours of type +4 and +6 are shown in Figure 5, which amount to at most eight neighbours of these types.Note that these can exist only if deg G (s) = 6.It remains to count the neighbours of type +2, which will turn out at exactly 2 deg G (s), independent of the specifics of G. Observe first that there can be two types of neighbours T ∈ N Gn (s) of type +2 distinguished by the T -degree of s, which is either two or four (cf. Figure 6).We shall say that these neighbours are of type +2 2 and +2 4 respectively.
In the following, an r-chain is an inclusion chain s ⊂ ∆ ⊂ T , where ∆ is an s-incident triangle in G, and T is a neighbour of s of type +2 r .The following information can be read from Figure 6: a neighbour of s of type +2 r can be extended to an r-chain in exactly n r ways (where n 2 = 1 and n 4 = 3).Likewise, an s-incident triangle can be extended to an r-chain in exactly n r ways as well.By double counting, we find that 1/n r times the number of r-chains equals both the number of s-incident triangles (which is exactly deg G (s)) and the number of neighbours of s of type +2 r .In conclusion, the number of neighbours of s of type +2 is exactly 2 deg G (s). Taking together all of the above, we count and if there is no G n -neighbour of type +6, then the maximal amount of 26 neighbours cannot have been attained, and deg Gn (s) = 26 as well.
It remains to show that if G contains arbitrarily large triangular-shaped subgraphs, then the graph invariant D(G n ) is both finite and unbounded as n → ∞.We first prove finitude of Proof.Suppose S ∼ = ∆ m with m ∈ 2N.We distinguish two cases.
Case 1: there is a T ∈ V (G n ) of side length µ ≥ 6 and dist Gn (S, T ) ≤ 2. We then fix a maximally long path T 0 T 1 . . .T in G n with T 0 := T and T i ∼ = ∆ µ+6i (i.e., T i and T i+1 are adjacent of type ±6; see Figure 7).Since the path is maximal, T has no G nneighbour of type +6, and since T is of side length µ + 6 ≥ µ ≥ 6, we have T ∈ deg 26 (G n ) by Lemma 3.1.As a vertex of G n , T is of side length at most n, and hence  Case 2: there is no T ∈ V (G n ) of side length µ ≥ 6 and dist Gn (S, T ) ≤ 2. Then we can conclude two things: first, m < 6 (otherwise, choose T := S) and so there is an s ∈ N Gn (S) of side length zero.Second, s has no neighbour of type +6 (otherwise, set T to be this neighbour).But then s cannot have degree 26 by Lemma 3.2, and therefore Finally, we show that D(G n ) is unbounded as n → ∞, assuming that there are arbitrarily large triangular-shaped subgraphs of G.
Proof.Choose a triangular-shaped graph S ∈ V (G n ) of side length n ∈ 48N.Roughly, the idea is to define a set M ⊆ deg 26 (G n ) that contains "deep vertices", i. e., vertices that have no "short" G n -paths that lead out of M. We claim that the following set has all the necessary properties: The following observation will be used repeatedly and we shall abbreviate it by ( * ): if  We first verify M ⊆ deg 26 (G n ).Fix T ∈ M and consider an embedding of S into the hexagonal lattice.In this embedding, T ⊆ S has a neighbour T of type +6 that, for all we know, might partially lie outside of S; though we now show that actually T ⊆ S: in fact, for all v ∈ V (T ) holds where we used both ( * ) and T ∈ M in the second inequality.Thus T ⊆ S and T also exists in G.Note that this argument shows that all G n -neighbours of T are contained in S. We denote the latter fact by ( * * ) as we reuse it below.For now we conclude that since T has a G n -neighbour of type +6, we have Next we identify a "deep vertex" in M, that is, a vertex with distance to V (G n ) \ M of more than n/48.We claim that we can choose for this the "central" triangular-shaped subgraph S ∼ = ∆ n/2 .By that we mean the triangular-shaped graph obtained from S by repeatedly deleting the boundary n/6 times.The resulting triangular-shaped subgraph has side length n/2 and dist G (S , ∂S) = n/6.Since n ≥ 48, we have both m 0 := n/2 ≥ 6 and dist G (S , ∂S) = n/6 ≥ 4, and therefore S ∈ M. It remains to show that we have := dist Gn (S , V (G n ) \ M) > n/48.Let S 0 . . .S be a path in G n from S 0 := S to some S ∈ M. Let m i ∈ N 0 be the side length of S i .Since S −1 ∈ M, by ( * * ) we have S ⊆ S. Thus, for S to be not in M, only two reasons are left, and we verify that either implies > n/48: • Case 1: m < 6.Since S i−1 and S i are adjacent in G n they can differ in side length by at most six (via an adjacency of type ±6).That is, m i−1 − m i ≤ 6, and thus It then follows In both cases, the right-most inequality was obtained using n ≥ 48.
Since in our setting we have where D( • ) is the graph invariant defined in (3.1).In particular, if G contains arbitrarily large triangular-shaped subgraphs, then D(k n G) is unbounded as n → ∞, and G is therefore clique divergent.
Together with [1, Theorem 7.9] we conclude the characterisation of clique convergent triangularly simply connected locally cyclic graphs of minimum degree δ ≥ 6.
Theorem A (Characterisation theorem for triangularly simply connected graphs).A triangularly simply connected locally cyclic graph of minimum degree δ ≥ 6 is clique divergent if and only if it contains arbitrarily large triangular-shaped subgraphs.

Proof of Theorem B
In this section we prove Theorem B. We need to recall basic facts about group actions and graph coverings, which we do in Section 4.1 and Section 4.2 below.

Group Actions, Γ-Isomorphisms, and Quotient Graphs
We say that a group Γ acts on a graph G if we have a group homomorphism σ : Γ → Aut(G).For every γ ∈ Γ and every v ∈ V (G), we define γv := σ(γ)(v).The graph G together with this action is called a Γ-graph.For every subgroup Γ ≤ Aut(G), G is a Γ-graph in a natural way.For two Γ-graphs G and H, we call a graph isomorphism If G is a Γ-graph, so is kG with respect to the induced action γQ = {γv | v ∈ Q}.Note that in [6] this action is denoted as the natural action of the group Γ k ≤ Aut(kG), which is isomorphic to Γ.For a second Γ-graph H and a Γ-isomorphism φ : G → H, the map We establish that our geometric construction of clique graphs is Γ-isomorphic as well.
Remark 4.2.If a Γ-graph G is locally cyclic, triangularly simply connected and of minimum degree δ ≥ 6, the action of Γ on G induces an action on the triangular-shaped subgraphs of G which makes the geometric clique graph G n into a Γ-graph as well.Furthermore, the isomorphism ψ n : G n → k n G provided by Theorem 2.2 is a Γ-isomorphism.This follows from the fact that the isomorphisms C n : G n → kG n−1 , which are explicitly constructed in [1, Corollary 6.9], are Γ-isomorphisms, and that ψ n can be written as the following chain of Γ-isomorphisms: For any vertex v ∈ V (G) of a Γ-graph G, we denote the orbit of v under the action of Γ by Γv.These orbits form the vertex set of the quotient graph G/Γ, two of which are adjacent if they contain adjacent representatives.Note that if two graphs G and H are Γ-isomorphic, the quotient graphs G/Γ and H/Γ are isomorphic.

Triangular Covers
In the following, we transfer the convergence criterion of Theorem A from the triangularly simply connected case to the general case using the triangular covering maps from [6].
We define the topologically inspired term of "triangular simple connectivity" via the concept of walk homotopy.As usual, a walk of length in a graph G is a finite sequence of vertices α = v 0 . . .v such that each pair v i−1 v i of consecutive vertices is adjacent.The vertex v 0 is called the start vertex, the vertex v is called the end vertex, a walk is called closed if start and end vertex coincide, and it is called trivial if it has length zero.
In order to define the homotopy relation on walks, we define four types of elementary moves (see also Figure 9).Given a walk that contains three consecutive vertices that form a triangle in G, the triangle removal shortens the walk by removing the middle  one of them.Conversely, if a walk contains two consecutive vertices that lie in a triangle of G, the triangle insertion lengthens the walk by inserting the third vertex of the triangle between the other two.The dead end removal shortens a walk that contains a vertex twice with distance two in the walk by removing one of the two occurrences as well as the vertex between them.Conversely, the dead end insertion lengthens a walk by inserting behind one vertex an adjacent one and then the vertex itself again.
Note that elementary moves do not change the start and end vertices of walks, not even of closed ones.
Two walks are called homotopic if it is possible to transform one into the other by performing a finite number of elementary moves.The graph G is called triangularly simply connected if it is connected and if every closed walk is homotopic to a trivial one.
A triangular covering map is a homomorphism p : G → G between two connected graphs which is a local isomorphism, i. e., the restriction p| N [ṽ] : N [ṽ] → N [p(ṽ)] to the closed neighbourhood of any vertex ṽ of G is an isomorphism and in this case, G is called a triangular cover of G.The term "triangular" refers to the unique triangle lifting property which can be used as an alternative definition and is defined in Appendix B. For a triangular covering map p : G → G, we define the map p k n : k n G → k n G which is constructed from p recursively by p k 0 = p and p k n ( Q) = {p k n−1 (ṽ) | ṽ ∈ Q} for n ≥ 1.By [6, Proposition 2.2], p k n is a triangular covering map, as well.
A triangular covering map p : G → G is called universal if G is triangularly simply connected, and in this case G is called the universal (triangular) cover of G.Note that every connected graph has a universal cover that is unique up to isomorphism.A proof can be found in [16,Theorem 3.6] or in the appendix in Theorem B.5.
For the following lemma, we need to use that triangular simple connectivity is preserved under the clique operator.This is proven in [10], but we also provide a elementary proof in the appendix in Lemma A.2. Lemma 4.3.If a connected graph G is clique convergent, so is its universal triangular cover G.
Proof.Let the clique operator be convergent on G, i. e., there are n, r ∈ N such that k n G ∼ = k n+r G, and let p : G → G be a universal triangular covering map.As p k n and p k n+r are triangular covering maps and k n G and k n+r G are triangularly simply connected by Lemma A.2, they are universal triangular covering maps.As the universal cover is unique up to isomorphism ( Theorem B.5), k n G ∼ = k n+r G and G is clique convergent.
In the following, we show that for locally cyclic graphs with minimum degree δ ≥ 6 the converse implication is true as well.This has been stated in [1] as Lemma 8.8, but the proof contains a gap, as it does not show that k n G and k n+r G are Γ-isomorphic (in fact, this is still unknown if G is a cover of a general graph G; see also Question 5.5).We will close this gap in the remainder of this section.
In order to do this, we need the definition of Galois maps.For a group Γ, we call a triangular covering map p : G → G Galois with Γ if G is a Γ-graph such that the vertex preimages of p are exactly the orbits of the action, which implies G/Γ ∼ = G.By [6, Proposition 3.2], if p is Galois with Γ, so is p k n .
The following lemma is proven in [1, Lemma 8.7], but again, an elementary proof is provided in Lemma B.6.We are now able to deduce the clique convergence of a graph from the clique convergence of its universal cover.Lemma 4.5.Let G be a locally cyclic graph with minimum degree δ ≥ 6 and G its universal triangular cover.If G is clique convergent, then so is G.
Proof.We start with the universal triangular cover G being clique convergent.By Theorem A, there is an m ∈ N such that G does not contain ∆ m as a subgraph.Consequently, Gm−2 and Gm are identical and thus Γ-isomorphic (for every Γ).
Let Γ be the deck transformation group of the universal covering map p : G → G.By Lemma 4.4, this implies k n G ∼ = (k n G)/Γ for each n ∈ N 0 .Using the Γ-isomorphism ψ n : k n G → Gn from Remark 4.2, we conclude that G is clique convergent via By joining Lemma 4.3, Lemma 4.5, and Theorem A, we conclude the characterisation of clique convergent locally cyclic graphs with minimum degree δ ≥ 6.
Theorem B (General characterisation theorem).A (not necessarily finite) connected locally cyclic graph of minimum degree δ ≥ 6 is clique divergent if and only if its universal triangular cover contains arbitrarily large triangular-shaped subgraphs.

Conclusion and Further Questions
In this article, we completed the characterisation of locally cyclic graphs of minimum degree δ ≥ 6 with a convergent clique dynamics, first in the triangularly simply connected case (Theorem A) and then in the general case (Theorem B).
Our findings turned out to be consistent with the geometric intuition from the finite case: the hexagonal lattice is clique divergent, as is any of its quotients.The finite analogues are the 6-regular triangulations of surfaces with Euler characteristic zero, which were known to be clique divergent by [5,6].We are tempted to say that the hexagonal lattice is clique divergent because it has a "flat geometry".
Theorem A may allow for a similar interpretation: if a triangularly simply connected locally cyclic graph G of minimum degree δ ≥ 6 is clique divergent, then it contains arbitrarily large triangular-shaped subgraphs.As a consequence, vertices of degree ≥ 7 cannot be distributed densely everywhere in G. Since degrees ≥ 7 can be interpreted as a discrete analogue of negative curvature (we think of the 7-regular triangulation of the hyperbolic plane), a potential geometric interpretation of Theorem A is that G is clique divergent because it is "close to being flat" on large parts, which then dominate the clique dynamics.
To consolidate this interpretation, it would be helpful to shed more light on the lower degree analogues: locally cyclic graphs of minimum degree δ = 5 or even δ = 4.There however, the clique dynamics might be governed by different effects.In a sense, it was surprising to find that for minimum degree δ ≥ 6, the asymptotic behaviour of the clique dynamics is determined only on the global scale, that is, by the presence or absence of subgraphs in G from a relatively simple infinite family (the triangular-shaped graphs).Such a description should not be expected for smaller minimum degree: for δ ≤ 5 there exist finite graphs that are clique divergent -even simply connected ones -and such clearly cannot contain "arbitrarily large" forbidden structures in any sense.
It might be worthwhile to first study triangulations of the plane of minimum degree δ = 5 or δ = 4, since those are not subject to the same argument of "finite size".Yet, as far as we are aware, it is already unknown which of the following graphs are clique divergent: consider a triangulated sphere of minimum degree δ ∈ {4, 5} (e.g. the octahedron or icosahedron).Remove a vertex or edge together with all incident triangles -which leaves us with a triangulated disc -and extend this to a triangulation of the Euclidean plane that is 7-regular outside the interior of the disc (see Figure 10).For all we know, it is at least conceivable that below minimum degree δ = 6 divergence can appear as a local phenomenon that does not require arbitrarily large "bad regions".
For triangulations of closed surfaces (and further mild assumptions, see below), the most elementary open question is whether non-negative Euler characteristic already implies clique-divergence.This has previously been conjecture by Larrión, Neumann-Lara and Pizaña [8], and we shall repeat it here.To shed further light on the perceived connection between topology and clique dynamics, the study of further topologically motivated generalisations appears worthwhile.We briefly mention two of them.
First, one could turn to higher-dimensional analogues, that is, triangulations of higherdimensional manifolds and their 1-skeletons.Question 5.2.Can something be said about when the clique dynamics of the triangulation of a manifold converges depending on the topology of the manifold?
The second generalisation is to allow for triangulations of surfaces with boundary.Such triangulations can be formalised as graphs for which each open neighbourhood is either a cycle (of length at least four) or a path graph -we shall call them locally cyclic with boundary.Triangulations of bordered surfaces have already received some attention: in [11,Theorem 1.4] the authors show that, except for the disc, each compact surface (potentially with boundary) admits a clique divergent triangulation.In contrast, they conjecture that discs do not have divergent triangulations: Conjecture 5.3.If a locally cyclic graph G with boundary and of minimum degree δ ≥ 4 triangulates a disc, then it is clique convergent (actually, clique null, that is, it converges to the one-vertex graph).This is known to be true if all interior vertices of the triangulation have degree ≥ 6 [7,Theorem 4.5].
Moving on from the topologically motivated investigations, yet another route is to generalise from locally cyclic graphs of a particular minimum degree to graphs of a lowerbounded local girth (that is, the girth of each open neighbourhood is bounded from below).In fact, it has already been noted by the authors of [8] that their results apply not only to locally cyclic graphs of minimum degree ≥ 7, but equally to general graphs of local girth ≥ 7. Various other open questions emerge from the context of graph coverings.As we have seen in Lemma 4.3, if a graph G is clique convergent, so is its universal triangular cover G.Even stronger: if k n G ∼ = G, then k n G ∼ = G.If G is locally cyclic of minimum degree δ ≥ 6, then conversely, by Lemma 4.5 convergence of G implies convergence of G.
For general triangular covers p : G → G (between connected locally finite graphs) however, such connections are not known.If both G and G are finite, then a straightforward pigeon hole argument shows that clique convergence of G and of G are equivalent.Yet, whether finite or infinite, it is generally unknown whether the statements k n G ∼ = G and k n G ∼ = G are always equivalent.We summarize all of this in the following question: Question 5.5.Let p : G → G be a triangular covering map between two connected locally finite graphs.Is G clique convergent if and only if G is clique convergent?To consider the directions separately, we ask: (i ) Is there an analogue of Lemma 4.3 for non-universal covering maps: if G is clique convergent but p is not universal, is G clique convergent as well?
(ii ) If G is clique convergent, is G clique convergent as well?
An even stronger version of the question is: is

A. The Clique Graph Operator and Simple Connectivity
In this section, we show that triangular simple connectivity is preserved under the clique graph operator.A weaker version was obtained by Prisner [15] in 1992, who proved that the clique graph operator preserves the first Z 2 Betti number.Larrión and Neumann-Lara [6] then extended this in 2000 to the isomorphism type of the triangular fundamental group.An extension to more general graph operators (including the clique graph operator and the line graph operator) was proven by Larrión, Pizaña, and Villarroel-Flores [10] in 2009.The proof presented here is completely elementary, as it explicitly constructs a sequence of elementary moves that transforms a given closed walk to the trivial one.
In order to be triangularly simply connected, the clique graph first needs to be connected.
Lemma A.1.For a connected graph G, the clique graph kG is also connected.
For each i ∈ {1, . . ., } we choose a clique Q i that contains the pair of consecutive vertices v i−1 and v i of this walk.Thus, for each i ∈ {1, . . ., − 1}, the cliques Q i and Q i+1 intersect in v i and they are distinct, as otherwise the vertices v i−1 and v i+1 would be adjacent, in contradiction to the minimality of the walk v 0 . . .v .Thus, The resulting walk connects Q and Q in kG and, thus, kG is connected.
We establish a concept of correspondence between a walk in G and a walk in kG in order to use the elementary moves that morph the former one into a trivial one as a guideline for doing the same with the latter one.We say that a closed walk α in G and a closed walk α = Q 0 . . .Q in kG with Q 0 = Q correspond if for each i ∈ {0, . . ., − 1} there is a walk v i,0 . . .v i,t i of length t i ∈ N 0 that lies completely in Q i and α is the concatenation of those walks, i. e., v i,t i = v i+1,0 for each i ∈ {0, . . ., − 1}.As α is closed, we the have v 0,0 = v −1,t −1 =: v ,0 .Note that for every closed walk in kG there is a corresponding one in G, which is obtained as follows.Let α = Q 0 . . .Q with Q 0 = Q be a closed walk in kG.For every i ∈ {1, . . ., }, we choose w i ∈ Q i−1 ∩ Q i , we define w 0 := w , and we drop repeated consecutive vertices.This way, we obtain a walk α which clearly corresponds to α .
Lemma A.2.If G is a triangularly simply connected graph, so is kG.
Proof.Let G be a triangularly simply connected graph.Thus, G is connected and, by Lemma A.1, so is kG.Next, we show that every closed walk in kG can be morphed to a single vertex by a sequence of elementary moves.Let α = Q 0 . . .Q with Q 0 = Q be a closed walk in kG.Let α be any corresponding walk in G, thus α consists of subwalks v i,0 . . .v i,t i as described above.
Since G is triangularly simply connected, there is a sequence of elementary moves from α to a trivial walk.We now describe how we use the first of these moves as a guideline for elementary moves on α , for the other moves in the sequence, it works by induction.
Let β be the walk in G that is reached from α by the first move.We now perform two steps in order to construct a walk β in kG, which is homotopic to α and which corresponds to β.The first step consists of repeated triangle removals and dead end removals on α that preserve the correspondence to α until α cannot be shortened any further in that way.
As no elementary move can change the start and end vertex of a walk, we do not remove Q 0 = Q this way.As for every i ∈ {1, . . ., − 1} with t i = 0, the clique Q i can be removed in a triangle or dead end removal, the only t i that can be zero is t 0 .For the second step, we distinguish two cases.Case 1: insertion moves.If the elementary move from α to β is a triangle insertion or dead end insertion, let the indices i ∈ {0, . . ., − 1} and j ∈ {0, . . ., t i − 1} be chosen such that the additional one or two vertices are inserted between v i,j and v i,j+1 .For the triangle insertion, the subwalk v i,0 . . .v i,t i becomes v i,0 . . .v i,j v * v i,j+1 . . .v i,t i and for the dead end insertion, it becomes v i,0 . . .v i,j v * v i,j v i,j+1 . . .v i,t i .If v * ∈ Q i , β := α corresponds to β and we are finished.If v * / ∈ Q i , let Q * be a clique that contains v * , v i,j and in the case of a triangle insertion also v i,j+1 .Then, the dead end inclusion of Q * and Q i behind Q i yields a walk β .In the case of a dead end inclusion, it corresponds to β because v i,0 . . .v i,j and v i,j . . .v i,t i lie in Q i and v i,j v * v i,j lies in Q * .In the case of a triangle inclusion, it corresponds to β because v i,0 . . .v i,j and v i,j+1 . . .v i,t i lie in Q i and v i,j v * v i,j+1 lies in Q * .Case 2: removal moves.If the elementary move from α to β is a triangle removal or dead end removal, let the indices i ∈ {0, . . ., − 1} and j ∈ {0, . . ., t i − 1} be chosen such that v i,j (triangle removal) or v i,j and v i,j+1 (dead end removal) are removed from Q i .This choice is possible, as the (first) removed vertex and its successor lie in a common Q i .If j ≥ 1, the walk β = α corresponds to β as v i,0 . . .v i,j−1 v i,j+1 . . .v i,t i or v i,0 . . .v i,j−1 v i,j+2 . . .v i,t i respectively, still lie in Q i .In case of a dead end removal, this works even if t i = j + 1, as then v i,j−1 = v i,j+1 = v i+1,0 .If j = 0, we know that i = 0, as otherwise v i,j = v 0,0 would be removed.Furthermore, we know that if i = 1, t 0 = 0 as this also would imply that v 0,0 = v 1,0 is removed.In any case, v i,j lies between v i−1,t i−1 −1 and v i,1 .We now distinguish between two cases.Case 2.1: and Q i is a triangle insertion and thus the resulting walk β is homotopic to α .Furthermore, β and β correspond, because Case 2.2: We start by assuming that v i,1 ∈ Q i−1 .We subdivide α differently in pieces that each lie in one clique Q i .Let t i−1 := t i−1 + 1, t i := t i − 1 and t s := t s for every s ∈ {0, . . ., − 1} \ {i − 1, i}.Furthermore, let for every u ∈ {0, . . ., t i }, and let v s,u := v s,u for every s ∈ {0, . . ., −1}\{i−1, i} and every u ∈ {0, . . ., t s }.Now, the removed vertex is v i−1,t i−1 and as t i−1 ≥ 1 we are in a case we have already treated.The step for v i−1,t i−1 −1 ∈ Q i is analogous.
After proceeding inductively for the other moves of the sequence, we reach a closed walk in kG which corresponds to a trivial walk in G. Thus, all vertices of that walk in kG are pairwise connected, as they all contain the single vertex of that trivial walk, and the walk can easily be morphed into a trivial one.

B. Some Background on (Universal) Triangular Covers
In this section, we provide some background on triangular covering maps.We start with some preliminaries on walk homotopy in the preimage and image of a triangular covering map.After that, we spend the main part of this section showing that the universal cover of a connected graph is unique up to isomorphism and covers every other triangular cover of a connected graph.Afterwards, we show that the universal covering map is Galois, i. e., that it can be interpreted as factoring out a group of symmetries from a graph.Most of the proofs are based on ideas from [16], but they only use basic concepts and they are much more concise as they use stronger prerequisites than the respective theorems in [16] have.We remark that every triangular covering map p : G → G fulfils the unique edge lifting property, i. e., for each pair of adjacent vertices v, w ∈ V (G) and each ṽ ∈ V ( G) such that p(ṽ) = v, there is a unique w ∈ V ( G) such that ṽ and w are adjacent and p( w) = w.This property is equivalent to the unique walk lifting property, which says that for each walk α in G and each preimage of its start vertex there is a unique walk α in G which is mapped to α.Furthermore, triangular covering maps fulfil the triangle lifting property, i. e., for each triangle (i.e. 3-cycle) {u, v, w} in G and each preimage ũ of u, there exists a unique triangle {ũ, ṽ, w} in G that is bijectively mapped to {u, v, w}.Lastly, it follows from the unique walk lifting property that every triangular covering map between two connected graphs is surjective.Throughout this section, we repeatedly make use of the following lemma connecting triangular covering maps and homotopy of walks.
Proof.As homotopy is defined by a finite sequence of elementary moves, it suffices to show that an elementary move in the image implies an elementary move in the preimage.Thus, let α = v 0 . . .v be a walk in G and let α = ṽ0 . . .ṽ be from its preimage with p(ṽ i ) = v i .Let β be reached from α by inserting a vertex v * and possibly v i again between v i and v i+1 for some i ∈ {0, . . ., − 1}.As lifting a walk is done vertex by vertex from start to end, the lift of β begins with the vertices ṽ0 to ṽi .As the restriction of p to the neighbourhood of v i−1 is an isomorphism, the lift of β starting in ṽ0 still has ṽi+1 as the preimage of v i+1 and consequently the lift of β agrees with that of α in all following vertices.Thus, the lift of β arises from the lift of α by inserting a vertex ṽ * and possibly ṽi between ṽi and ṽi+1 , which is an elementary move.For the elementary moves that remove vertices, exchange α and β.
Next, we show that every connected graph has a universal triangular cover.The proof of the following lemma is influenced by a combination of [16, Theorem 2.5, 2.8, and 3.6].
Lemma B.2.For every connected graph G, there is a universal triangular covering map p : G → G, i. e., a triangular covering map with a triangularly simply connected graph G.
Proof.We give a construction for a graph G and a map p and we show that p is in fact a triangular covering map, that G is connected and that G is triangularly simply connected.Construction of G and p: We fix a vertex v of G.For each walk α, we denote by [α] its homotopy class, i. e., the set of walks that can be reached from α by a finite sequence of elementary moves.A walk β is called a continuation of a walk α if β arises from α by appending exactly one vertex to its end.Now we can define the graph G by in which end(α) is the end vertex of α.The map p is well defined as homotopic walks have the same start and end vertex.Triangular covering map: For an edge [α][β] ∈ E( G), let without loss of generality β be a continuation of α.Thus, the end vertices of the two walks are adjacent and p is a graph homomorphism.Next we show that the restriction of p to neighbourhoods is bijective.Thus, let [α w ] be a class of walks from v to some vertex w.As noted above, the neighbourhood of [α w ] consists of the classes of continuations of α w to the neighbours of w.Especially, the restriction of p to the neighbourhoods of [α w ] and w respectively is bijective.Let now be α x and α y be the continuations of α w by two distinct neighbours x and y of w.As we have already shown that the adjacency of [α x ] and [α y ] implies the adjacency of x and y, it remains to show the reverse.Thus, let x and y be adjacent.Hence, we can construct the walk α y as the continuation of α x by the vertex y, thus, [α x ] and [α y ] are adjacent.Since α y is reached from α y by he elementary move of inserting x between w and y, they are homotopic and thus also [ we can assume without loss of generality that α i is a continuation of α i−1 for each i ∈ {1, . . ., }.Furthermore, we can assume that α 0 is the trivial walk as all the walks α 0 , . . ., α coincide with α 0 on their initial subwalks, anyway.We prove that the closed [α 0 ] . . .[α ] and the trivial walk [α 0 ] are homotopic.As α 0 and α are homotopic, there is a finite sequence of elementary moves that morphs α into α 0 .To each walk α in G that occurs in that homotopy between α 0 and α , we associate the walk [α 0 ] . . .[α ] where is the length of α and α i is the initial subwalk of length i of α .This is a walk by construction and it fulfils α 0 = α 0 and α = α .This way, we associate the final (trivial) walk α 0 to the trivial walk [α 0 ].If the walks α and α are connected by an elementary move in G, their associated walks in G are connected by the corresponding elementary move in the following way.A triangle insertion move that inserts v * after v i corresponds to the insertion of the class of the continuation of α i by v * and changing the representative of the following classes to the one, in which v * is inserted after v i .The other elementary moves work analogously.
In the next lemma, we show that universal triangular covers are in fact universal objects.The proof is a combination of special cases from the proofs of [16, Theorem 3. Lemma B.3.The universal triangular covering map p : G → G fulfils the following universal property: for each triangular covering map q : Ḡ → G there exists a triangular covering map q : G → Ḡ such that p = q • q (see the commuting diagram in Figure 14).Furthermore, for any pair of fixed vertices ṽ ∈ G and v ∈ Ḡ such that p(ṽ) = q(v) we get a unique triangular covering map qṽ,v : G → Ḡ with p = q • qṽ,v and qṽ,v (ṽ) = v.Proof.Let p : G → G be a triangular covering map such that G is triangularly simply connected and let q : Ḡ → G be any triangular covering map.We fix a vertex v ∈ V (G) as well as vertices ṽ ∈ V ( G) and v ∈ V ( Ḡ) that are in the preimage of v under p and q, respectively.We construct qṽ,v from p and q and show that it is in fact a well-defined triangular covering map.Construction of qṽ,v : For each ũ ∈ V ( G), we choose a walk α ṽ,ũ from ṽ to ũ.The image of α ṽ,ũ under p is a walk, which we call β ũ, from p(ṽ) to p(ũ).As p(ṽ) = v = q(v), by the unique walk lifting property, there is exactly one walk α v,ū starting at v that is mapped to β ũ by q.We define qṽ,v (ũ) to be the end vertex ū of α v,ū .Well-Definedness: We need to show that qṽ,v (ũ) is independent of the choice of the walk α ṽ,ũ .Thus, let α ṽ,ũ be a different walk from ṽ to ũ.Its image under p is called β ũ which has the same start and end vertices as β ũ.As G is triangularly simply connected, the walks α ṽ,ũ and α ṽ,ũ are homotopic and, consequently, so are β ũ and β ũ.By Lemma B.1 also the preimages under q, which are called α v,ū and α v,ū , are homotopic and, thus, have the same end vertex, implying qṽ,v being well defined.Additionally, p = q • qṽ,v holds by construction.
Homomorphy: Let x, ỹ be adjacent vertices in G. Let α ṽ,ỹ be a walk from ṽ to ỹ such that x is its penultimate vertex.Via the same construction as above, we obtain a walk α v,ȳ such that its penultimate vertex x fulfils p(x) = q(x).Consequently, qṽ,v (x) = x and qṽ,v (ỹ) = ȳ are adjacent and thus qṽ,v is a graph homomorphism.Triangular covering map: Let ũ be a vertex of G and let u = p(ũ) and ū = qṽ,v (ũ) be its images.As .Uniqueness of qṽ,v : Let q : G → Ḡ be any triangular covering map such that p = q • q and q(ṽ) = v.With the definitions from above, both the image of α ṽ,ũ under q and α v,ū are lifts of the walk β ũ and they share the start vertex v.By the unique walk lifting property, they are equal and so is their end vertex, implying q(ũ) = ū = qṽ,v (ũ).
Lemma B.4.If for a graph G there are two graphs G and Ḡ and two triangular covering maps p : G → G and q : Ḡ → G such that p and q both fulfil the universal property from Lemma B.3, G and Ḡ are isomorphic.
Proof.Let p : G → G and q : Ḡ → G be two triangular covering maps which both fulfil the universal property.Furthermore, let ṽ ∈ V ( G) and v ∈ V ( Ḡ) be chosen such that p(ṽ) = q(v).By the universal properties, there are (unique) triangular covering maps p : Ḡ → G and q : G → Ḡ such that p = q • q, q(ṽ) = v, q = p • p, and p(v) = ṽ.Consequently, p = p • p • q and (p • q)(ṽ) = ṽ.As the identity map id : G → G is a triangular covering map that fulfils p = p • id and id(ṽ) = ṽ, we know by the uniqueness of the universal property of p that p • q = id, which implies that q : G → Ḡ is an isomorphism.
Theorem B.5.Every connected graph has a universal triangular cover, which is unique up to isomorphism.
Proof.By Lemma B.2, the graph G has a universal triangular cover.Let p : G → G and q : Ḡ → G be two universal triangular covering maps.By applying Lemma B.3, they both have the universal property.By Lemma B.4, the universal triangular covers are isomorphic.Now we can look at the universal triangular cover through the lens of quotient graphs by using Galois covering maps.We reprove this lemma from [1] using only basic notation.Proof.As each γ ∈ Γ fulfils p • γ = p, the group Γ acts on every vertex preimage of p individually.Thus, it suffices to show that for each pair of vertices ṽ, w with p(ṽ) = p( w) there is a γ ∈ Γ such that γ(ṽ) = w.If we apply Lemma B.3 with q = p, we get a triangular covering map qṽ, w which maps ṽ to w and which is an isomorphism by Theorem B.5, thus γ = qṽ, w fulfils the condition.As p is a Galois covering map, by [6, Proposition 3.2] so is p k n .Consequently, it holds that (k n G)/Γ ∼ = k n G.

Figure 3 .
Figure 3.: The 26 possible ways in which a triangular-shaped graph S ∈ V (G n ) of side length m ≥ 6 can be G n -adjacent to another triangular-shaped graph T ∈ V (G n ) of side length m + s, where s ∈ {−6, −4, −2, 0, +2, +4, +6}.Two configurations may differ merely by a symmetry (one of the six "reflections" and "rotations" of a triangularshaped 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 ±2, ±4 and ±6, 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 • (3 + 3 + 3 + 1).

Figure 5 .:
Figure 5.: The eight possible neighbours of a triangular-shaped graph of side length m = 0 of type +4 and +6.See the caption of Figure 3 for an explanation of the multiplicities.

Figure 6 .
Figure 6.: Row +2 r shows the ways in which an inclusion s ⊂ T (left; T being a G n -neighbour of s of type +2 r ) or an inclusion s ⊂ ∆ (right; ∆ being an triangle in G) extends to an r-chain in n r = r − 1 ways.

Figure 7 .
Figure 7.: Initial segment T 0 T 1 T 2 . . . of an increasing path of triangular-shaped subgraphs of Gwhere T i and T i+1 are adjacent of type ±6.

Lemma 3 . 4 .
If G contains a triangular-shaped subgraph of side length n ∈ 48N, then there exists an S for all v ∈ T .This can be verified by considering the configurations shown in Figure3.The bound ≤ 4 is best possible as seen in Figure8.

Figure 8 .
Figure 8.: The "corner vertex" v of T ∈ V (G n ) (light grey) has G-distance four to the neighbour T ∈ N Gn (T ) of type −6 (dark grey).

Conjecture 5 . 1 .
If a locally cyclic graph G of minimum degree δ ≥ 4 triangulates a closed surface of Euler characteristic χ ≥ 0 (i.e., a sphere, projective plane, torus or Klein bottle), then G is clique divergent.

Figure 10 .
Figure 10.: An "almost 7-regular" triangulation of the Euclidean plane, that is, it is 7-regular outside a small region.

Question 5 . 4 .
Can the results for locally cyclic graphs of minimum degree δ ≥ 6 be generalized to graphs of local girth ≥ 6?

Figure 11 .:
Figure 11.: The correspondence relation between a walk in G and one in kG.

Figure 12 .:
Figure 12.: The elementary move in kG that corresponds to a dead end insertion (left) or triangle insertion (right) of a vertex which is not in Q i .

Figure 13 .:
Figure 13.: The elementary move in kG that corresponds to triangle removal in G.
α y ] is adjacent to [α x ].Connectivity: We show that every vertex [α] is connected to the trivial walk α v that consists only of the vertex v. Thus, let α be any walk in G.The vertices [α v ] and [α] are connected by the walk [β 0 ] . . .[β ] in G, where is the length of α and β i is the initial subwalk of length i of α.Triangular simple connectivity: For a closed walk [α 0

Lemma B. 6 .
A universal triangular covering map p : G → G is Galois with Γ := {γ ∈ Aut( G) | p • γ = p}, which is called the deck transformation group of p.Moreover, it holds that (k n G)/Γ ∼ = k n G.