A Lower Bound on the Diameter of the Flip Graph

The flip graph is the graph whose nodes correspond to non-isomorphic combinatorial triangulations and whose edges connect pairs of triangulations that can be obtained one from the other by flipping a single edge. In this note we show that the diameter of the flip graph is at least $\frac{7n}{3} + \Theta(1)$, improving upon the previous $2n + \Theta(1)$ lower bound.


Introduction
A combinatorial triangulation is a maximal planar graph (a planar graph to which no edge can be added without destroying planarity) together with a clockwise ordering for the edges incident to each vertex. An intuitive way to define a combinatorial triangulation is as an equivalence class of planar drawings (say on the sphere) of a maximal planar graph, where two drawings are equivalent if a continuous morph exists from one drawing to the other that does not create crossings or overlaps between edges. We are interested in simple combinatorial triangulations, which have no self-loops or multiple edges. In the following, when we say triangulation we always mean simple combinatorial triangulation. Observe that, in a planar drawing equivalent to a triangulation, all the faces are delimited by cycles with three vertices (hence the name triangulation).
Consider a planar drawing Γ on the sphere equivalent to a triangulation G and consider an edge (a, b) in G. If (a, b) were removed from Γ , there would exist a unique region of the sphere delimited by a cycle with four edges; in fact the cycle delimiting such region would be (a, a ′ , b, b ′ ), for some vertices a ′ and b ′ . The operation of flipping (a, b) consists of removing (a, b) from G and inserting the edge (a ′ , b ′ ) inside the region delimited by the cycle (a, a ′ , b, b ′ ). The resulting triangulation G ′ might not be simple though. In the following, we only refer to flips that maintain the triangulations simple.
The flip graph G n describes the possibility of transforming n-vertex triangulations using flips. The vertex set of G n is the set of distinct n-vertex triangulations; two triangulations G and H are connected by an edge in G n if there exists an edge e of G such that flipping e in G results in H.
Various properties of the flip graph have been studied. A particular attention has been devoted to the diameter of G n , which is the length of the longest (among all pairs of vertices) shortest path; refer to the surveys [3,5]. A first proof that the diameter of G n is finite goes back to almost a century ago [11]. A sequence of deep improvements [4,[7][8][9][10] have led to the current best upper bound of 5n + Θ(1), which was proved this year by Cardinal et al. [7]. Significantly less results and techniques have been presented for the lower bound. We are only aware of a 2n + Θ(1) lower bound on the diameter of G n , which was proved by Komuro [8] by exploiting the existence of triangulations with "very different" vertex degrees. The main contribution of this note is the following theorem.
Theorem 1. For every n ≥ 3, the diameter of the flip graph is at least 7n 3 − 34.

Proof of the Main Result
In this section we prove Theorem 1. Let n ≥ 3. For a triangulation G, we denote by V (G) and E(G) its vertex and edge set, respectively. Consider any n-vertex triangulation G 1 . A path incident to G 1 in G n is a sequence of n-vertex triangulations such that the first triangulation in the sequence is G 1 and any two triangulations which are consecutive in the sequence can be obtained one from the other by flipping a single edge. Thus, a path incident to G 1 in G n corresponds to a valid sequence σ = (u 1 , v 1 ), . . . , (u k , v k ) of flips, where u 1 , . . . , u k , v 1 , . . . , v k are vertices in V (G 1 ) and (u i , v i ) is an edge of the triangulation obtained starting from G 1 by performing flips (u 1 , v 1 ), . . . , (u i−1 , v i−1 ) in this order. For a valid sequence σ of flips, denote by G σ 1 the n-vertex triangulation obtained starting from G 1 by performing the flips in σ. Observe that V (G 1 ) = V (G σ 1 ), given that a flip only modifies the edge set of a triangulation, and not its vertex set. Now consider any two n-vertex triangulations G 1 and G 2 and consider a simple path in G n between them. This path corresponds to a valid sequence σ of flips transforming G 1 into G 2 . By the definition of G n , the n-vertex triangulations G σ 1 and G 2 are isomorphic; that is, there exists a bijective mapping γ : The key idea for the proof of Theorem 1 is to consider the bijective mapping γ before the flips in σ are applied to G 1 and to derive a lower bound on the number of flips in σ based on properties of γ. In fact, the property we employ is the number of common edges of G 1 and G 2 according to γ.
More precisely, for a bijective mapping γ : V (G 1 ) → V (G 2 ) between the vertex sets of two triangulations G 1 and G 2 , we define the number c γ of common edges with respect to γ as the number of distinct edges (u, v) ∈ E(G 1 ) such that (γ(u), γ(v)) ∈ E(G 2 ). We have the following. Lemma 1. For any two n-vertex triangulations G 1 and G 2 , the number of flips needed to transform G 1 into G 2 is at least 3n−6−max γ c γ , where the maximum is over all bijective mappings γ : Proof. The statement descends from the following two observations. First, two isomorphic n-vertex triangulations have 3n−6 common edges according to the bijective mapping γ realizing the isomorphism. Second, for any two n-vertex triangulations H and L that have ℓ common edges with respect to any bijective mapping γ, flipping any edge in H results in a combinatorial triangulation H ′ such that H ′ and L have at most ℓ + 1 common edges with respect to γ.
It remains to define two n-vertex triangulations G 1 and G 2 such that any bijective mapping γ between their vertex sets has a small number c γ of common edges.
-Triangulation G 1 is defined as follows (see Fig. 1a). Let H be any triangulation of maximum degree six with ⌊ n 3 ⌋ + 2 vertices. Note that the number of faces of H is 2(⌊ n 3 ⌋ + 2) − 4 = 2⌊ n 3 ⌋. If n ≡ 2 modulo 3, if n ≡ 1 modulo 3, or if n ≡ 0 modulo 3, then insert a vertex inside each face of H, insert a vertex inside each face of H except for one face, or insert a vertex inside each face of H except for two faces, respectively. When a vertex is inserted inside a face of H, it is connected to the three vertices of H incident to the face. Denote by G 1 the resulting n-vertex triangulation. We say that the vertices of G 1 in H are blue, while the other vertices of G 1 are red.
-Triangulation G 2 is defined as follows (see Fig. 1b). Starting from a path P with n − 2 vertices, connect all the vertices of P to two further vertices a and b, and connect a with b.
We have the following.
Lemma 2. For any bijective mapping γ : V (G 1 ) → V (G 2 ), we have c γ ≤ 2⌊ n 3 ⌋ + 28. Proof. Consider any bijective mapping γ : V (G 1 ) → V (G 2 ). First, note that each vertex v ∈ V (G 1 ) has degree at most twelve. Namely, v has at most six blue neighbors; further, v has at most six incident faces in H, hence it has at most six red neighbors. It follows that, whichever vertex in V (G 1 ) is mapped to a according to γ, at most twelve out of the n − 1 edges incident to a are shared by G 1 and G 2 with respect to γ. Analogously, at most twelve out of the n − 1 edges incident to b are shared by G 1 and G 2 with respect to γ. It remains to bound the number of edges of P that are shared by G 1 and G 2 with respect to γ. This proof uses a pretty standard technique (see, e.g., [6,7]). Since G 1 has no edge connecting two red vertices, the number of edges of P that are shared by G 1 and G 2 with respect to γ is at most the number of edges of P that have at least one of their end-vertices mapped to a blue vertex; since ⌊ n 3 ⌋ + 2 vertices of G 1 are blue, there are at most 2⌊ n 3 ⌋ + 4 such edges of P . It follows that the number of edges shared by G 1 and G 2 with respect to γ is at most 2⌊ n 3 ⌋ + 28. By Lemma 2, we have that G 1 and G 2 are two n-vertex triangulations such that, for any bijective mapping γ : V (G 1 ) → V (G 2 ), we have c γ ≤ 2⌊ n 3 ⌋ + 28. By Lemma 1, the number of flips needed to transform G 1 into G 2 is at least 3n − 6 − 2⌊ n 3 ⌋ − 28 ≥ 7n 3 − 34. This concludes the proof of Theorem 1.

Conclusions
In this note we have presented a lower bound of 7n 3 + Θ(1) on the diameter of the flip graph for n-vertex triangulations. One of the main ingredients for this lower bound is a lemma stating that there exist two n-vertex triangulations such that any bijective mapping γ between their vertex sets creates at most c γ ≤ 2n 3 + Θ(1) common edges. It not clear to us whether the bound resulting from this approach can be improved further. That is, is it true that, for every two n-vertex triangulations, there exists a bijective mapping γ between their vertex sets creating c γ ≥ 2n 3 + Θ(1) common edges? The only lower bound on the value of c γ we are aware of comes as a corollary of the fact that every n-vertex triangulation has a matching of size at least n+4 3 as proved in [2], hence c γ ≥ n+4 3 . It is an interesting fact that, for every n-vertex triangulation H, a bijective mapping γ : V (H) → V (G 2 ) exists creating c γ = 2n 3 + Θ(1) common edges, where G 2 is the graph from the proof of Theorem 1. In fact, every n-vertex triangulation H has a set of n 3 + Θ(1) vertex-disjoint simple paths covering its vertex set V (H), as proved by Barnette [1] (this bound is the smallest possible [6]). Mapping these paths to sub-paths of the path P in G 2 provides the desired bijective mapping γ.