Crooked Graphs: Theory and Applications

• Crooked graphs are specialized graph constructions derived from crooked functions over F₂ that yield distance-regular, triangle-free, and antipodal graphs.
• In geometry, similar structures arise as crooked planes, halfspaces, and surfaces, though these objects are distinct from the graph-theoretic definition.
• In interval dynamics, the term describes graphs of interval maps with specific crookedness conditions that are crucial for characterizing pseudo-arc inverse limits.

“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 (Eberhard et al., 27 Aug 2025, Činč, 2024, Charette et al., 2010).

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 $K_{2,t}$" (Eberhard et al., 27 Aug 2025), 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 (Eberhard et al., 27 Aug 2025).

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 (Charette et al., 2010, Burelle et al., 2012). In the Einstein universe, the relevant objects are crooked surfaces, which are conformal compactifications of crooked planes (Charette et al., 2013, Burelle et al., 2017). In the anti-de Sitter setting, the objects are AdS crooked planes (Danciger et al., 2014). 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 (Charette et al., 2015).

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" (Činč, 2024) 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 (Činč, 2024).

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	$Q:V\to W$0	Graph of an interval map whose iterates are crooked

2. Crooked functions and crooked graphs over $Q:V\to W$1

Let $Q:V\to W$2 and $Q:V\to W$3 be $Q:V\to W$4-dimensional vector spaces over $Q:V\to W$5. A function

$Q:V\to W$6

is crooked if:

1. $Q:V\to W$7,
2. $Q:V\to W$8 for all distinct $Q:V\to W$9 with

$F_2$0

1. $F_2$1 for all $F_2$2 and all $F_2$3 (Eberhard et al., 27 Aug 2025).

The prototypical example recorded in the paper is

$F_2$4

when $F_2$5 with $F_2$6 odd (Eberhard et al., 27 Aug 2025).

Given a crooked function $F_2$7, the associated crooked graph $F_2$8 has vertex set

$F_2$9

Distinct vertices $\Gamma(f)$0 and $\Gamma(f)$1 are adjacent iff

$\Gamma(f)$2

This is the paper’s exact adjacency rule (Eberhard et al., 27 Aug 2025).

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 $\Gamma(f)$3, but it records that the graph $\Gamma(f)$4 can also serve as a recognition device for the function-theoretic notion (Eberhard et al., 27 Aug 2025).

3. Structural properties and use in diameter-$\Gamma(f)$5 extremal graph constructions

The paper imports the following properties of $\Gamma(f)$6 from Bending and Fon-Der-Flaass and from de Caen, Mathon, and Moorhouse:

1. $\Gamma(f)$7 is odd.
2. $\Gamma(f)$8 is distance-regular of order $\Gamma(f)$9, degree $K_{2,t}$0, and diameter $K_{2,t}$1.
3. $K_{2,t}$2 is triangle-free.
4. Any pair of vertices at distance two have exactly two common neighbors.
5. $K_{2,t}$3 is antipodal.
6. The map

$K_{2,t}$4

defines a $K_{2,t}$5-fold cover

$K_{2,t}$6

whose fibres $K_{2,t}$7 are the cliques of the distance-$K_{2,t}$8 graph and such that the edges between any two distinct fibers form a perfect matching (Eberhard et al., 27 Aug 2025).

These imported properties are the entire reason crooked graphs are useful in the paper. Triangle-freeness and the “exactly two common neighbors at distance $K_{2,t}$9” condition control local codegrees; antipodality and the fibre decomposition isolate the unique obstruction to diameter $$W_t=\{\text{graphs other than stars that have diameter }2\text{ and contain neither a triangle nor a }K_{2,t}\}.$$0, namely the pairs lying in the same fibre (Eberhard et al., 27 Aug 2025).

For $$W_t=\{\text{graphs other than stars that have diameter }2\text{ and contain neither a triangle nor a }K_{2,t}\}.$$1, the construction starts with $$W_t=\{\text{graphs other than stars that have diameter }2\text{ and contain neither a triangle nor a }K_{2,t}\}.$$2 and a crooked function $$W_t=\{\text{graphs other than stars that have diameter }2\text{ and contain neither a triangle nor a }K_{2,t}\}.$$3, for example $$W_t=\{\text{graphs other than stars that have diameter }2\text{ and contain neither a triangle nor a }K_{2,t}\}.$$4. The authors adjoin vertices

$$W_t=\{\text{graphs other than stars that have diameter }2\text{ and contain neither a triangle nor a }K_{2,t}\}.$$5

and define

$$W_t=\{\text{graphs other than stars that have diameter }2\text{ and contain neither a triangle nor a }K_{2,t}\}.$$6

The resulting graph $$W_t=\{\text{graphs other than stars that have diameter }2\text{ and contain neither a triangle nor a }K_{2,t}\}.$$7 has order

$$W_t=\{\text{graphs other than stars that have diameter }2\text{ and contain neither a triangle nor a }K_{2,t}\}.$$8

is triangle-free, $$W_t=\{\text{graphs other than stars that have diameter }2\text{ and contain neither a triangle nor a }K_{2,t}\}.$$9-free, and has diameter $Q(x)=x^{2^k+1}$0. Hence

$Q(x)=x^{2^k+1}$1

The paper emphasizes that these examples are not regular (Eberhard et al., 27 Aug 2025).

For $Q(x)=x^{2^k+1}$2, the construction is recursive. If $Q(x)=x^{2^k+1}$3 has order $Q(x)=x^{2^k+1}$4, then one embeds a copy of $Q(x)=x^{2^k+1}$5 into each fibre of a crooked graph $Q(x)=x^{2^k+1}$6 on $Q(x)=x^{2^k+1}$7 vertices. The resulting graph $Q(x)=x^{2^k+1}$8 remains triangle-free and $Q(x)=x^{2^k+1}$9-free and has diameter $\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\}$0. Theorem 2 states: $\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\}$1 Thus crooked graphs yield an infinite family of regular members of $\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\}$2 (Eberhard et al., 27 Aug 2025).

The same paper contrasts these crooked-graph constructions with a separate $\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\}$3 family built from Cayley graphs on $\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\}$4. That contrast is technically important: the $\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\}$5 and $\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\}$6 results are genuinely crooked-graph constructions, whereas the $\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\}$7 examples come from a different algebraic mechanism (Eberhard et al., 27 Aug 2025).

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 $\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\}$8-space is a piecewise-linear surface built from a central stem and two wings (Charette et al., 2010). In the notation of Charette and Goldman, the model crooked plane $\Gamma(f)=\{(x,f(x))\in I\times I:x\in I\}$9 is

$G_Q$0

and for a general unit spacelike vector $G_Q$1 and vertex $G_Q$2,

$G_Q$3

(Charette et al., 2010).

Burelle, Charette, Drumm, and Goldman developed the corresponding crooked halfspaces $G_Q$4, with boundary

$G_Q$5

and proved that every point in an open crooked halfspace lies on a particle contained in it (Burelle et al., 2012). 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 $G_Q$6-space preserves the partial order defined by inclusion and the involution defined by complementarity (Burelle et al., 2012).

Frances’ Einstein-universe theory replaces crooked planes by crooked surfaces. A crooked surface is an element in the $G_Q$7-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 (Charette et al., 2013). The same paper proves two basic topological facts: a crooked surface is homeomorphic to a Klein bottle, and a crooked surface separates $G_Q$8 (Charette et al., 2013).

In the symplectic model of $G_Q$9, 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 (Burelle et al., 2017). 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 $V\times F_2\times W$0-space by crooked planes. For a normalized path of pairwise ultraparallel spacelike directors $V\times F_2\times W$1 and a regular curve $V\times F_2\times W$2, the family $V\times F_2\times W$3 is a crooked foliation if and only if

$V\times F_2\times W$4

for all $V\times F_2\times W$5, where

$V\times F_2\times W$6

is the stem quadrant (Charette et al., 2013).

Danciger, Guéritaud, and Kassel strengthened this picture by proving that any two disjoint crooked planes in $V\times F_2\times W$7 are leaves of a crooked foliation: $V\times F_2\times W$8 Equivalently, every pair of disjoint crooked planes occurs as two leaves of some smooth foliation by pairwise disjoint crooked planes (Burelle et al., 2019).

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 $V\times F_2\times W$9 by the arc complex $Q:V\to W$00 via a homeomorphism

$Q:V\to W$01

and used this parameterization to prove the Crooked Plane Conjecture for convex cocompact Margulis spacetimes (Danciger et al., 2014). 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 (Charette et al., 2015). 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 (Danciger et al., 2014).

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

In interval dynamics, the relevant graph is literal: $Q:V\to W$02 Here a map $Q:V\to W$03 is $Q:V\to W$04-crooked between $Q:V\to W$05 and $Q:V\to W$06 if for every $Q:V\to W$07 with

$Q:V\to W$08

there is a point $Q:V\to W$09 between $Q:V\to W$10 and $Q:V\to W$11, and there is a point $Q:V\to W$12 between $Q:V\to W$13 and $Q:V\to W$14, such that

$Q:V\to W$15

The map is $Q:V\to W$16-crooked if it is $Q:V\to W$17-crooked between every pair of points from $Q:V\to W$18, and it is crooked if for every $Q:V\to W$19 there exists $Q:V\to W$20 such that $Q:V\to W$21 is $Q:V\to W$22-crooked (Činč, 2024).

The decisive criterion is: $Q:V\to W$23 Thus, in this setting, crookedness is exactly the graph-folding condition on a bonding map whose inverse limit is the pseudo-arc (Činč, 2024).

The paper then specializes to maps under or above the identity diagonal. It defines “under diagonal” by

$Q:V\to W$24

and “above diagonal” by

$Q:V\to W$25

A key restriction is: $Q:V\to W$26 This shows that a crooked under-diagonal map cannot be piecewise monotone (Činč, 2024).

Henderson’s map is the prototype. It is described as starting with $Q:V\to W$27 and perturbing its graph with infinitely many $Q:V\to W$28-shaped notches which accumulate in a particular fashion on the point $Q:V\to W$29 (Činč, 2024). 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 (Činč, 2024).

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 $Q:V\to W$30 and $Q:V\to W$31 constructions rely on preexisting properties of crooked graphs; the paper’s novelty lies in modifying those graphs to force diameter $Q:V\to W$32, not in redefining or reclassifying crooked graphs themselves (Eberhard et al., 27 Aug 2025).

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 (Charette et al., 2010). "Fundamental domains in the Einstein Universe" likewise states that the relevant term is crooked surfaces, not crooked graphs (Charette et al., 2013). "Einstein tori and crooked surfaces" studies crooked surfaces and Einstein tori, not graphs (Burelle et al., 2017).

There are also setting-specific limitations. The foliation theorem for disjoint crooked planes is specific to $Q:V\to W$33-dimensional Minkowski space and does not address higher-dimensional analogues (Burelle et al., 2019). The interval-dynamical theory is specifically about interval maps; it does not provide a theory of crooked maps on arbitrary topological graphs (Činč, 2024). In anti-de Sitter geometry, some proper actions admit no crooked fundamental domain at all (Danciger et al., 2014).

A common source of confusion is therefore terminological rather than mathematical. The cited papers support three distinct uses: a finite graph $Q:V\to W$34 derived from a crooked function; the graph $Q:V\to W$35 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.