---
title: 'Crooked Graphs: Theory and Applications'
url: https://www.emergentmind.com/topics/crooked-graphs
type: topic
---

# Crooked Graphs: Theory and Applications

“Crooked graphs” does not denote a single standard object across the arXiv literature. In finite graph theory, the term refers to the graph \(G_Q\) associated to a crooked function \(Q:V\to W\) over \(F_2\). In interval dynamics, the natural reading is the graph \(\Gamma(f)\) of a crooked interval map. By contrast, several papers in affine Lorentzian, Einstein, and anti-de Sitter geometry explicitly state that they do **not** define a formal notion called a crooked graph; there the mathematically correct objects are crooked planes, crooked halfspaces, crooked surfaces, crooked foliations, crooked ideal triangles, and related combinatorial structures built from them [2508.19646] [2405.20533] [1005.1315].

## 1. Terminology, scope, and historical placement

The graph-theoretic usage is the most literal one. In "Examples of diameter-2 graphs with no triangle or \(K_{2,t}\)" [2508.19646], a crooked graph is defined from a crooked function and is used to construct infinite families in the classes
\[
W_t=\{\text{graphs other than stars that have diameter }2\text{ and contain neither a triangle nor a }K_{2,t}\}.
\]
The paper places these graphs in a historical line running from de Caen, Mathon, and Moorhouse to Bending and Fon-Der-Flaass. It states that crooked graphs were first named by Bending and Fon-Der-Flaass after abstracting the key properties of the crooked functions \(Q(x)=x^{2^k+1}\) considered by de Caen, Mathon, and Moorhouse [2508.19646].

Several geometric papers explicitly warn that “crooked graph” is not their terminology. In affine Minkowski space, the basic objects are crooked planes, crooked halfspaces, crooked polyhedra, crooked tilings, and zigzags [1005.1315] [1211.4177]. In the Einstein universe, the relevant objects are crooked surfaces, which are conformal compactifications of crooked planes [1307.6531] [1702.08414]. In the anti-de Sitter setting, the objects are AdS crooked planes [1410.5804]. In the Margulis-spacetime literature, the phrase may informally point to graph- and cell-like structures built from crooked planes, such as the trivalent tree of superbases and the tile decomposition of deformation space [1501.04535].

In interval dynamics, “crooked graphs” refers naturally to the graphs of interval maps whose iterates satisfy a precise crookedness condition. The paper "On zero entropy homeomorphisms of the pseudo-arc" [2405.20533] makes this interpretation explicit by writing
\[
\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\}
\]
and by treating crookedness as the graph-theoretic/dynamical condition characterizing pseudo-arc inverse limits [2405.20533].

| Context | Object | Defining description |
|---|---|---|
| Finite graph theory | \(G_Q\) | Graph on \(V\times F_2\times W\) defined from a crooked function |
| Affine/Lorentzian geometry | Crooked plane or halfspace | Piecewise object built from a stem and two wings |
| Einstein/AdS geometry | Crooked surface or AdS crooked plane | Compactified or curved analogue of crooked planes |
| Interval dynamics | \(\Gamma(f)\) | Graph of an interval map whose iterates are crooked |

## 2. Crooked functions and crooked graphs over \(F_2\)

Let \(V\) and \(W\) be \(n\)-dimensional vector spaces over \(F_2\). A function
\[
Q:V\to W
\]
is crooked if:

1. \(Q(0)=0\),
2. \(\sum_{i=1}^4 Q(x_i)\ne 0\) for all distinct \(x_1,x_2,x_3,x_4\in V\) with
   \[
   x_1+x_2+x_3+x_4=0,
   \]
3. \(\sum_{i=1}^3\bigl(Q(x_i)+Q(x_i+a)\bigr)\ne 0\) for all \(x_1,x_2,x_3\in V\) and all \(a\in V\setminus\{0\}\) [2508.19646].

The prototypical example recorded in the paper is
\[
Q(x)=x^3
\]
when \(V=W=F_{2^n}\) with \(n\) odd [2508.19646].

Given a crooked function \(Q\), the associated crooked graph \(G_Q\) has vertex set
\[
V\times F_2\times W.
\]
Distinct vertices \((a,i,\alpha)\) and \((b,j,\beta)\) are adjacent iff
\[
\alpha+\beta = Q(a+b) + (i+j+1)(Q(a)+Q(b)).
\]
This is the paper’s exact adjacency rule [2508.19646].

The same paper notes, in a footnote, that Godsil and Roy characterized crooked functions in terms of the distance-regularity of the corresponding graph defined by this adjacency relation. That remark is historical and structural: the paper itself defines crookedness via \(Q\), but it records that the graph \(G_Q\) can also serve as a recognition device for the function-theoretic notion [2508.19646].

## 3. Structural properties and use in diameter-\(2\) extremal graph constructions

The paper imports the following properties of \(G_Q\) from Bending and Fon-Der-Flaass and from de Caen, Mathon, and Moorhouse:

1. \(n\) is odd.
2. \(G_Q\) is distance-regular of order \(2q^2\), degree \(2q-1\), and diameter \(3\).
3. \(G_Q\) is triangle-free.
4. Any pair of vertices at distance two have exactly two common neighbors.
5. \(G_Q\) is antipodal.
6. The map
   \[
   (a,i,\alpha)\mapsto (a,i)
   \]
   defines a \(q\)-fold cover
   \[
   G_Q\to K_{2q}
   \]
   whose fibres \(I_1,\dots,I_{2q}\) are the cliques of the distance-\(3\) graph and such that the edges between any two distinct fibers form a perfect matching [2508.19646].

These imported properties are the entire reason crooked graphs are useful in the paper. Triangle-freeness and the “exactly two common neighbors at distance \(2\)” condition control local codegrees; antipodality and the fibre decomposition isolate the unique obstruction to diameter \(2\), namely the pairs lying in the same fibre [2508.19646].

For \(W_3\), the construction starts with \(q=2^{2e-1}\) and a crooked function \(Q:F_q\to F_q\), for example \(Q(x)=x^3\). The authors adjoin vertices
\[
v_1,\dots,v_{2q},v
\]
and define
\[
N(v_j)=I_j\cup\{v\},\qquad N(v)=\{v_1,\dots,v_{2q}\}.
\]
The resulting graph \(G'_Q\) has order
\[
2q^2+2q+1=2^{4e-1}+2^{2e}+1,
\]
is triangle-free, \(K_{2,3}\)-free, and has diameter \(2\). Hence
\[
G'_Q\in W_3.
\]
The paper emphasizes that these examples are not regular [2508.19646].

For \(W_5\), the construction is recursive. If \(H\in W_5\) has order \(q\), then one embeds a copy of \(H\) into each fibre of a crooked graph \(G_Q\) on \(2q^2\) vertices. The resulting graph \(G''_Q\) remains triangle-free and \(K_{2,5}\)-free and has diameter \(2\). Theorem 2 states:
\[
\text{For each integer } e\ge 2 \text{ there is a regular graph } G\in W_5 \text{ of order } 2^{2^e-1}.
\]
Thus crooked graphs yield an infinite family of regular members of \(W_5\) [2508.19646].

The same paper contrasts these crooked-graph constructions with a separate \(W_7\) family built from Cayley graphs on \(F_p^2\). That contrast is technically important: the \(W_3\) and \(W_5\) results are genuinely crooked-graph constructions, whereas the \(W_7\) examples come from a different algebraic mechanism [2508.19646].

## 4. Crooked planes, crooked halfspaces, and crooked surfaces as nearby but distinct notions

In the affine Lorentzian literature, the nearest objects to a “crooked graph” are crooked planes and crooked halfspaces. A crooked plane in Minkowski \(2+1\)-space is a piecewise-linear surface built from a central stem and two wings [1005.1315]. In the notation of Charette and Goldman, the model crooked plane \(\mathcal C_0\) is
\[
\mathcal C_0=\mathcal W_0^- \cup S_0 \cup \mathcal W_0^+,
\]
and for a general unit spacelike vector \(\mathbf v\) and vertex \(p\in\mathbb E\),
\[
\mathcal C(\mathbf v,p)=p+\mathcal C(\mathbf v)
\]
[1005.1315].

Burelle, Charette, Drumm, and Goldman developed the corresponding crooked halfspaces \(H(p,s)\), with boundary
\[
\partial H(p,s)=C(p,s),
\]
and proved that every point in an open crooked halfspace lies on a particle contained in it [1211.4177]. They also showed that the set of parallelism classes of timelike lines, or particles, in a crooked halfspace is a geodesic halfplane in the hyperbolic plane, and that the correspondence between crooked halfspaces and halfplanes in hyperbolic \(2\)-space preserves the partial order defined by inclusion and the involution defined by complementarity [1211.4177].

Frances’ Einstein-universe theory replaces crooked planes by crooked surfaces. A crooked surface is an element in the \(\mathrm{SO}(3,2)\)-orbit of the conformal compactification of a crooked plane; it is built from a stem lying in an Einstein torus and two wings that are half-lightcones [1307.6531]. The same paper proves two basic topological facts: a crooked surface is homeomorphic to a Klein bottle, and a crooked surface separates \(Ein^3\) [1307.6531].

In the symplectic model of \(\Ein^3\), the crooked surface is again a three-piece object—two wings and one stem—and the disjointness problem is reduced to boundary photons. Two crooked surfaces are disjoint if and only if the four photons on the boundary of the stem of one are disjoint from the other, and conversely [1702.08414]. This makes the terminology precise: in these geometric papers the correct objects are crooked planes, halfspaces, or surfaces, not graphs.

## 5. Foliations, deformation theory, and graph-like combinatorics built from crooked planes

A second family of near-matches arises from crooked foliations and from combinatorial structures indexed by arcs, superbases, or tiles. Charette and Kim gave an infinitesimal criterion for foliations of Minkowski \(2+1\)-space by crooked planes. For a normalized path of pairwise ultraparallel spacelike directors \(u_t\) and a regular curve \(p_t\), the family \((p_t,u_t)\) is a crooked foliation if and only if
\[
\dot p_t\in V(u_t)
\]
for all \(t\), where
\[
V(u)=\{a\,u^- - b\,u^+ : a,b\ge 0\}\setminus\{0\}
\]
is the stem quadrant [1312.6674].

Danciger, Guéritaud, and Kassel strengthened this picture by proving that any two disjoint crooked planes in \(R^3\) are leaves of a crooked foliation:
\[
\text{Let }C,C' \text{ be a pair of disjoint crooked planes in }R^3.\text{ Then there is a crooked foliation } C_t,\ 0\le t\le 1,\text{ with }C_0=C,\ C_1=C'.
\]
Equivalently, every pair of disjoint crooked planes occurs as two leaves of some smooth foliation by pairwise disjoint crooked planes [1903.01587].

In the Margulis-spacetime literature, the phrase “crooked graphs” is sometimes a plausible shorthand for graph- and cell-like structures built from crooked planes, but the papers retain different formal terminology. Danciger, Guéritaud, and Kassel parameterized the projectivized admissible cone \(adm(\rho)\) by the arc complex \(X\) via a homeomorphism
\[
f:X\longrightarrow adm(\rho),
\]
and used this parameterization to prove the Crooked Plane Conjecture for convex cocompact Margulis spacetimes [1407.5422]. Charette, Goldman, Jones, and Stambaugh developed the one-holed torus case in terms of a trivalent tree of superbases, a tiling of proper affine deformation space by triangular tiles, and a decomposition of Margulis spacetimes by crooked ideal triangles and crooked ideal quadrilaterals [1501.04535]. A plausible implication is that, in these papers, “crooked graph” can only be understood indirectly through the combinatorics of arcs, tiles, and adjacency graphs associated with crooked-plane decompositions.

The anti-de Sitter counterpart is more restrictive. AdS crooked planes are genuine analogues of Drumm’s Minkowski crooked planes, but some proper actions do not admit fundamental domains bounded by pairwise disjoint AdS crooked planes, in contrast to the Minkowski setting [1410.5804].

## 6. Crookedness of interval-map graphs and the pseudo-arc

In interval dynamics, the relevant graph is literal:
\[
\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\},
\qquad I=[0,1].
\]
Here a map \(f\in C(I)\) is \(\delta\)-crooked between \(a\) and \(b\) if for every \(c,d\in I\) with
\[
f(c)=a,\qquad f(d)=b,
\]
there is a point \(c'\) between \(c\) and \(d\), and there is a point \(d'\) between \(c'\) and \(d\), such that
\[
|b-f(c')|<\delta
\quad\text{and}\quad
|a-f(d')|<\delta.
\]
The map is \(\delta\)-crooked if it is \(\delta\)-crooked between every pair of points from \(I\), and it is crooked if for every \(\delta>0\) there exists \(n\in\mathbb N\) such that \(f^n\) is \(\delta\)-crooked [2405.20533].

The decisive criterion is:
\[
\hat I_f \text{ is the pseudo-arc if and only if } f \text{ is crooked.}
\]
Thus, in this setting, crookedness is exactly the graph-folding condition on a bonding map whose inverse limit is the pseudo-arc [2405.20533].

The paper then specializes to maps under or above the identity diagonal. It defines “under diagonal” by
\[
f(x)\le x \quad \text{for every } x\in [0,1],
\]
and “above diagonal” by
\[
f(x)\ge x \quad \text{for every } x\in [0,1].
\]
A key restriction is:
\[
\text{If } f:I\to I \text{ is under diagonal and piecewise monotone, then } \hat I_f \text{ is homeomorphic to an arc.}
\]
This shows that a crooked under-diagonal map cannot be piecewise monotone [2405.20533].

Henderson’s map is the prototype. It is described as starting with \(f(x)=x^2\) and perturbing its graph with infinitely many \(v\)-shaped notches which accumulate in a particular fashion on the point \((1,1)\subset I\times I\) [2405.20533]. The same paper proves the existence of uncountably many pairwise non-conjugate zero entropy crooked interval maps with different sets of fixed points, and also uncountably many pairwise non-conjugate zero entropy crooked maps with exactly two fixed points [2405.20533].

## 7. Significance, limitations, and recurrent misconceptions

The graph-theoretic literature treats crooked graphs as a mature imported structure rather than a newly developed theory. The \(W_3\) and \(W_5\) constructions rely on preexisting properties of crooked graphs; the paper’s novelty lies in modifying those graphs to force diameter \(2\), not in redefining or reclassifying crooked graphs themselves [2508.19646].

The geometric literature repeatedly corrects the terminology. "Affine Schottky Groups and Crooked Tilings" states that the paper does not introduce an object under the name “crooked graph,” and points instead to crooked planes, crooked halfspaces, crooked polyhedra, crooked tilings, and zigzags [1005.1315]. "Fundamental domains in the Einstein Universe" likewise states that the relevant term is crooked surfaces, not crooked graphs [1307.6531]. "Einstein tori and crooked surfaces" studies crooked surfaces and Einstein tori, not graphs [1702.08414].

There are also setting-specific limitations. The foliation theorem for disjoint crooked planes is specific to \(3\)-dimensional Minkowski space and does not address higher-dimensional analogues [1903.01587]. The interval-dynamical theory is specifically about interval maps; it does not provide a theory of crooked maps on arbitrary topological graphs [2405.20533]. In anti-de Sitter geometry, some proper actions admit no crooked fundamental domain at all [1410.5804].

A common source of confusion is therefore terminological rather than mathematical. The cited papers support three distinct uses: a finite graph \(G_Q\) derived from a crooked function; the graph \(\Gamma(f)\) of a crooked interval map; and several geometric families of crooked planes or surfaces whose combinatorics may be graph-like but are not formally called crooked graphs. The term is exact in the first two settings and inexact in the third.

Source: https://www.emergentmind.com/topics/crooked-graphs