Graph Derangements

We introduce the notion of a graph derangement, which naturally interpolates between perfect matchings and Hamiltonian cycles. We give a necessary and sufficient condition for the existence of graph derangements on a locally finite graph. This result was first proved by W.T. Tutte in 1953 by applying some deeper results on digraphs. We give a new, simple proof which amounts to a reduction to the (Menger-Egervary-Konig-)Hall(-Hall) Theorem on transversals of set systems. Finally, we consider the problem of classifying all cycle types of graph derangements on m x n checkerboard graphs. Our presentation does not assume any prior knowledge in graph theory or combinatorics: all definitions and proofs of needed theorems are given.


The Cockroach Problem.
Consider a square kitchen floor tiled by 25 = 5 × 5 square tiles in the usual manner. Late one night the house's owner comes down to discover that the floor is crawling with cockroaches: in fact each square tile contains a single cockroach. Displeased, she goes to the kitchen cabinet and pulls out an enormous can of roach spray. The roaches sense what is coming and start skittering. Each roach has enough time to skitter from its own tile to any adjacent tile. But it will not be so good for two (or more) roaches to skitter to the same tile: that will make an obvious target. Is it indeed possible for the roaches to perform a collective skitter in this way? This is a nice problem to give to undergraduates: it is concrete, fun, and far away from what they think they should be doing in a math class.

Solution.
Suppose the tiles are painted black and white with a checkerboard pattern, and that the center square is black, so that there are 13 black squares and 12 white squares. Therefore there are 13 roaches who start out on black squares and are seeking a home on only 12 white squares. It is not possible -no more so than for pigeons! -for all 13 cockroaches to end up on different white squares.

A problem for mathematicians.
For a grown mathematician (or even an old hand at mathematical brainteasers) this is not a very challenging problem, since the above parity considerations will quickly leap to mind. Nevertheless there is something about it that encourages further contemplation. There were several other mathematicians at the Normal Bar and they were paying attention too. "What about the 6 × 6 case?" one of them asked. "It reminds me of the Brouwer fixed point theorem," muttered another. 1 One natural followup is to ask what happens for cockroaches on an m × n rectangular grid. The preceding argument works when m and n are both odd. On the other hand, if e.g. m = n = 2 it clearly is possible for the cockroaches to skitter, and already there are several different ways. For instance, we could divide the rectangle into two dominos and have the roaches on each domino simply exchange places. Or we could simply have them proceed in a (counter)clockwise cycle.
Dominos are a good idea in general: if one of m and n is even, then an m × n rectangular grid may be tiled with dominos, and this gives a way for the roaches to skitter. Just as above though this feels not completely satisfactory and one naturally looks for other skittering patterns: e.g. when m = n = 4 one can have the roaches in the inner 2 × 2 square skittering clockwise as before and then roaches in the outer ring of the square skittering around in a cycle: isn't that more fun? There are many other skittering patterns as well.
I found considerations like the above to be rather interesting (and I will come back to them later), but for me the real problem was a bit more meta: what is the mathematical structure underlying the Cockroach Problem, and what is the general question being asked about this structure?
Here we translate the Cockroach Problem into graph-theoretic terms. In so doing we get a graph theoretic problem which in a precise sense interpolates between two famous and classic problems: existence of perfect matchings and existence of Hamiltonian cycles. On the other hand, the more general problem does not seem to be well known. But it's interesting, and we present it here: it is the existence and classification of graph derangements.

Basic definitions.
Let G = (V, E) be a simple, undirected graph: that is, we are given a finite set V of vertices and a set E of edges, which are unordered pairs of distinct elements of V . For v 1 , v 2 ∈ V , we say that v 1 and v 2 are adjacent if {v 1 , v 2 } ∈ E and write v 1 ∼ v 2 . In other words, for a set V , to give a graph with vertex set V is equivalent to giving an anti-reflexive, symmetric binary relation on V , the adjacency relation. A variant formalism is also useful: we may think of a graph as a pair of sets (V, E) and an incidence relation on V × E. Namely, for x ∈ V and e ∈ E, x is incident to e if x ∈ e, or, less formally, if x is one of the two vertices comprising the endpoints of e. If one knows the incidence relation as a subset of V × E then one knows in particular for each e ∈ E the pair of vertices {v 1 , v 2 } which are incident to E and thus one knows the graph G.
For G = (V, E) and v ∈ V , the degree of v is the number of edges which are incident to v. A degree zero vertex is isolated; a degree one vertex is pendant.
A graph is finite if its vertex set is finite (and hence its edge set is finite as well). A graph is locally finite if every vertex has finite degree.
If G = (V, E) and G ′ = (V, E ′ ) are graphs with E ′ ⊂ E, we say G ′ is an edge subgraph of G: it has the same underlying vertex set as G and is obtained from G by removing some edges. If G = (V, E) and G ′ = (V ′ , E ′ ) are finite graphs with V ′ ⊂ V and E ′ the set of all elements of E linking two vertices in V ′ , we say G ′ is an induced subgraph of G.
Example 2.2: More generally, for n ∈ Z + and m 1 , . . . , m n ∈ Z + we may define an n-dimensional analogue R m1,...,mn of R m,n . Its vertex set is {1, . . . , m 1 } × . . . × {1, . . . , m n } and we decree that (x 1 , . . . , Example 2.3: For m ≥ 2, n ≥ 1 we define the Möbius checkerboard gaph M m,n . This is a graph with the same vertex set as R m,n and having edge set consisting of all the edges of R m,n together with (x, n) ∼ (x, 1) for all 1 ≤ x ≤ m. We put C m = M m,1 , the cycle graph. The checkerboard graph R m,n is an edge subgraph of the Möbius checkerboard graph M m,n .
Example 2.4: For m, n ≥ 2 we define the torus checkerboard graph T m,n . Again it has the same vertex set as R m,n and contains all of the edges of R m,n together with the following ones: for all 1 ≤ x ≤ m, (x, n) ∼ (x, 1) and for all 1 ≤ y ≤ n, (m, y) ∼ (1, y). The Möbius checkerboard graph M m,n is an edge subgraph of the torus checkerboard graph T m,n .
For a graph G = (V, E) and x ∈ V , we define the neighborhood of x as N x = {y ∈ V | x ∼ y}. More generally, for any subset X ⊂ V we define the neighborhood of X as N (X) = {y ∈ V | ∃x ∈ X such that x ∼ y} = x∈X N x .
Remark 2.5: Although x / ∈ N x , X and N (X) need not be disjoint. In fact, X ∩ N (X) = ∅ iff X is an independent set. Now the "cockroach skitterings" that we were asking about on R m,n can be enunciated in terms of any graph as follows.
Let Der G be the set of all graph derangements of G.
It is natural to also consider a slightly more general definition.
For a graph G = (V, E), a graph permutation of G is an injection f : Let Perm G be the set of all graph permutations of G. Thus a graph derangement is precisely a graph permutation which is fixed point free: Proof. Left to the reader as an exercise to get comfortable with the definitions.
Remark 2.6: If G ′ is an edge subgraph of G, any graph derangement (resp. graph permutation) σ of G ′ is also a graph derangement (resp. graph permutation) of G.

Cycles and surjectivity.
Given a graph G, we wish not only to decide whether Der G is nonempty but also to study its structure. The collection of all derangements on a given graph is likely to be a very complicated object: consider for instance Der K n , which has size asymptotic to n! e . Just as in the case of ordinary permutations and derangements, it seems interesting to study the possible cycle types of graph derangements and graph permutations on a given graph G. Let us give careful definitions of these.
First, let V be a set and f : V → V an function. For m ∈ Z + , let f m = f • . . . • f be the mth iterate of f . We introduce a relation ≈ on V as follows: for x, y ∈ V , x ≈ y iff there are m, n ∈ Z + such that f m x = f n y. This is an equivalence relation on V : the reflexivity and the symmetry are immediate, and as for the transitivity: if x, y, z ∈ V are such that x ≈ y and y ≈ z then there are a, b, c, d ∈ Z + with f a x = f b y and f c y = f d z, and then Now suppose f is injective: we now call the ≈-equivalence classes cycles. Let x ∈ V , and denote the cycle containing x by C x . Then: • C x is singly infinite iff it is infinite and there are y ∈ V , m ∈ Z + such that f m y = x and y is not in the image of f .
• C x is doubly infinite iff it is infinite and every y ∈ C x lies in the image of f . These cases are mutually exclusive and exhaustive, so f is surjective iff there are no singly infinite cycles. Suppose G = (V, E) is a graph and f ∈ Perm G. We can define the cycle type of f as a map from the set of possible cycle types into the class of cardinal numbers. When V is finite, this amounts to a partition of #V in the usual sense: e.g. the cycle type of roaches skittering counterclockwise on a 5 × 5 grid is (1, 8, 16). A graph permutation is a graph derangement if it has no 1-cycles. We will say a graph derangement is matchless if it has no 2-cycles.
Proposition 2. Suppose that a graph G admits a graph derangement. Then G admits a surjective graph derangement.
Proof. It is sufficient to show that any graph derangement can be modified to yield a graph derangement with no singly infinite cycles, and for that it suffices to consider one singly infinite cycle, which may be viewed as the derangement n → n + 1 on the graph Z + with n ∼ n + 1 for all n ∈ Z + . This derangement can be decomposed into an infinite union of 2-cycles: 1 ↔ 2, 3 ↔ 4, . . . , 2n − 1 ↔ 2n, . . ..

Disconnected Graphs.
The proof is immediate. Thus we may as well restrict attention to connected graphs.

Bipartite Graphs.
A bipartition of a graph G = (V, E) is a partition V = V 1 V 2 of the vertex set such that each V i is an independent set. A graph is bipartite if it admits at least one bipartition.
There is a bijective correspondence between 2-colorings of G and bipartitions of G: given a 2-coloring C It follows that a graph is bipartite iff all of its connected components are bipartite.
Remark 2.8: Any subgraph G ′ of a bipartite graph G is bipartite. Indeed, any 2-coloring of G restricts to a 2-coloring of G ′ .
Example 2.9: The cycle graph C m is bipartite iff m is even.
Corollary 4. Let G be a bipartite graph, and let σ ∈ DerG. Then every finite cycle of σ has even length.
Proof. Since a subgraph of a bipartite graph is bipartite, a bipartite graph cannot admit a cycle of odd finite degree. 2 2.5. Some Examples.
Example 2.10: If G = K n is the complete graph on the vertex set [n] = {1, . . . , n}, then a graph permutation of G is nothing else than a permutation of [n]. A derangement of K n is a derangement in the usual sense, i.e., a fixed-point free permutation. Derangements exist iff n > 1 and the number of them is asymptotic to n! e as n → ∞.
The main tool in all of our Existence Theorems is a truly basic result of combinatorial theory. There are several (in fact, notoriously many) equivalent versions, but for our purposes it will be helpful to single out two different formulations.
We will deduce Theorem 5 from the following reformulation.
Proof. [HV50] Step 1: Suppose V 1 is finite. We go by induction on #V . The case #V 1 = 1 is trivial. Now suppose that #V 1 = n > 1 and that the result holds for all bipartitioned graphs with first vertex set of cardinality smaller than n. It will be notationally convenient to suppose that V 1 = {1, . . . , n}, and we do so. Case 1: Suppose that for all 1 ≤ k < n, every k-element subset of V 1 has at least k + 1 neighbors. Then we may match n to any element of V 2 and semiperfectly match {1, . . . , n − 1} into the remaining elements of V 2 by induction. Case 2: Otherwise, for some k, 1 ≤ k < n, there is a k-element subset X ⊂ V 1 such that #N (X) = k. The subset X may be semiperfectly matched into V 2 by induction, say via ι 1 : X → V 2 , so it suffices to show that the Hall Condition still holds on the induced bipartitioned subgraph on (V 1 \ X, V 2 \ ι 1 (X)). Indeed, if not, then for some h, 1 ≤ h ≤ n − k, there would be an h-element subset Y ⊂ V 1 \ X having fewer than h neighbors in V 2 \ ι 1 (X)), but then Step 2: Suppose V 1 is infinite. For x ∈ V 1 , endow N x with the discrete topology; endow N = x∈V1 N x with the product topology. Each N x is finite hence compact, so N is compact by Tychonoff's Theorem. For any finite subset X ⊂ V 1 , let Then H X is closed in N and is nonempty by Step 1. Since G is compact, there is n ∈ X H X , and any such n is a semiuperfect matching.
Remark 3.1: Theorems 5 and 6 are equivalent results: Assume Theorem 5. In the setting of Theorem 6 take V = V 2 , I = V 1 and S = {N x } x∈I . The local finiteness of the graph means each element of S is finite, and the assumed Cockroach Condition is precisely the Hall Condition, so by Theorem 5 there is X ⊂ V 2 and a bijection f : Let ι = f −1 : V 1 → X. Then for y ∈ V 1 , let y = ι(x), so x = f (y) ∼ y = ι(x).
Assume Theorem 6. In the setting of Theorem 5 take V 1 = I, V 2 = V , and E the set of pairs (i, x) such that x ∈ S i . Since each S i is finite, the graph is locally finite, and the assumed Hall Condition is precisely the Cockroach Condition, so by Theorem 6 there is a semiperfect matching ι : I → V . Let X = f (I), and let f : Remark 3.2: Theorem 5 was first proved for finite I by Philip Hall [Ha35]. Eventually it was realized that equivalent or stronger versions of P. Hall's Theorem had been proven earlier by Menger [Me27], Egerváry [Eg31] and Kőnig [Kő31]. The matrimonial interpretation was introduced some years later by Halmos and Vaughan [HV50]. Nevertheless, with typical disregard for history the most common name for the finite form of either Theorem 5 or 6 is Hall's Marriage Theorem.
Remark 3.3: The generalization to arbitrary index sets was given by Marshall Hall, Jr. [Ha48], whence "Halls' Theorem" (i.e., the theorem of more than one Hall).
Remark 3.4: M. Hall, Jr.'s argument used Zorn's Lemma, which is equivalent to the Axiom of Choice (AC). The proof supplied above uses Tychonoff's Theorem, which is also equivalent to (AC) [Ke50]. However, all of our spaces are Hausdorff. By examining the proof of Tychonoff's Theorem using ultrafilters [Ca37], one sees that when the spaces are Hausdorff, by the uniqueness of limits one does not need (AC) but only that every filter can be extended to an ultrafilter (UL). In turn, (UL) is equivalent to the fact that every Boolean ring has a prime ideal (BPIT). (BPIT) is known to be weaker than (AC), hence Halls' Theorem cannot imply (AC).

Question 1. Does Halls' Theorem imply the Boolean Prime Ideal Theorem?
Remark 3.5: The use of compactness of a product of finite, discrete spaces is a clue for the cognoscenti that it should be possible to find a nontopological proof using the Compactness Theorem from model theory. The reader who is interested and knowledgeable about such things will enjoy doing so. The Compactness Theorem (and also the Completeness Theorem) is known be equivalent to (BPIT).
Example 3.6 [EW49, pp. 288-289]: Let V 1 be the set of non-negative integers and let V 2 be the set of positive integers. For all positive integers x ∈ V 1 , we decree that x is adjacent to the corresponding positive integer in V 2 and to no other elements of V 2 . However, we decree that 0 ∈ V 1 is adjacent to every element of Y . It is clear that there is no semiperfect matching, because if we match 0 to any n ∈ V 2 , then the corresponding element n ∈ V 1 cannot be matched. But the Cockroach Condition holds: for a finite subset X ⊂ V 1 , if 0 = X then #N (V 1 ) = #V 1 , whereas if 0 ∈ X then N (V 1 ) = V 2 .
3.2. The First Existence Theorem.
Remark 3.7: Example 3.3 shows (H) need not imply (D) without the assumption of local finiteness. The graph with vertex set R and such that every x ∈ R is adjacent to every integer n > x satisfies (H ′ ) but not (H).

Lemma 8. Let G be a locally finite graph which violates the cockroach condition: there is a finite subset
Proof. Let Y ⊂ X be the subset of all vertices which are not adjacent to any element of X, so Y is an independent set. Put m 1 = #Y , m 2 = #(X \ Y ), and n = m 1 + m 2 = #X. By hypothesis, #N (X) Combining Proposition 2, Theorem 7 and Lemma 8 we deduce the following result. Remark 3.8: Theorem 9 was first proved by W.T. Tutte [Tu53, 7.1]. Most of Tutte's paper is concerned with related -but deeper -results on digraphs. The result which is our Theorem 9 appears at the end of the paper and is proven by passage to an auxiliary digraph and reduction to previous results. Perhaps because this was not the main focus of [Tu53], Theorem 9 seems not to be well known. In particular, our observation that one need only apply Theorem 5 appears to be new!

Bipartite Existence Theorems.
A matching on a graph V = (G, E) is a subset M ⊂ E such that no two edges in M share a common vertex. A matching M is perfect if every vertex of G is incident to exactly one edge in M.
A graph permutation is dyadic if all of its cycles have length at most 2.

Proposition 10. Let G be a graph. a) Matchings of G correspond bijectively to dyadic graph permutations. b) Under this bijection perfect matchings correspond to dyadic graph derangements.
Proof. a) Let M ⊂ E be a matching. We define σ M ∈ Sym V as follows: if x incident to the edge e = {x, y}, we put σx = y. Otherwise we put σx = x. This is well-defined since by definition every vertex is incident to at most one edge and gives rise to a dyadic graph permutation. Conversely, to any dyadic graph permutation σ ∈ Sym V , let X ⊂ V be the subset of vertices which are not fixed by σ and put M σ = x∈X {x, σx}. Then M → σ M and M → M σ are mutually inverse. b) A matching M is perfect iff every vertex x ∈ V is incident to an edge of M iff the permutation σ M is fixed-point free.
Let G = (V 1 , V 2 , E) be a bipartitioned graph. A semiperfect matching on G is a matching M ⊂ E such that every vertex of V 1 is incident to exactly one element of M. Thus a subset M ⊂ G is a perfect matching on (V 1 ∪V 2 , E) iff it is a semiperfect matching on both (V 1 , V 2 , E) and on (V 2 , V 1 , E).
But in fact we have defined the same thing twice: a semiderangement of a bipartitioned graph is nothing else than a semiperfect matching.

Then (SD) =⇒ (H), and if G is locally finite, (H) =⇒ (SD).
Proof. This is precisely Theorem 6 stated in the language of semiderangements.
Theorem 12. (Kőnigs' Theorem) Let G = (V 1 , V 2 , E) be a bipartitioned graph, which need not be locally finite. Suppose there is a semiderangement ι 1 : V 1 → V 2 of (V 1 , V 2 , E) and a semiderangement ι 2 : V 2 → V 1 of (V 2 , V 1 , E). Then G admits a perfect matching, i.e., a bijection f : Proof. Let V = V 1 V 2 . Then ι = ι 1 ι 2 : V → V is a graph derangement. The injection ι partitions V into finite cycles, doubly infinite cycles, and singly infinite cycles. For each cycle C which is finite or doubly infinite, ι 1 : C ∩ V 1 → C ∩ V 2 and ι 2 : C ∩V 2 → C ∩V 1 are bijections. Thus if there are no singly infinite cycles, taking f = ι 1 we are done. If C is a singly infinite cycle, it has an initial vertex x 1 . If x 1 ∈ V 1 , then ι 1 : C ∩ V 1 → C ∩ V 2 is surjective; the problem occurs if x 1 ∈ V 2 : then x 1 does not lie in the image of ι 1 . We have x 2 = ι 2 (x 1 ) ∈ V 1 , x 3 = ι 1 (x 2 ) ∈ V 2 , and so forth. So we can repair matters by defining ι 1 on x 2 , x 4 , . . . by x 2 → x 1 , x 4 → x 3 , and so forth. Doing this on every singly infinite cycle with initial vertex lying in V 2 a bijection f : V 1 → V 2 . Moreover, since ι 2 is a semiderangement and ι(x 2n−1 ) = x 2n for all n ∈ Z + , we have f (x 2n ) = x 2n−1 ∼ x 2n , so f is a bijection. Remark 3.9: Suppose V 1 and V 2 are sets and ι 1 : V 1 → V 2 and ι 2 : V 2 → V 1 are injections between them. If we apply Theorem 12 to the bipartitioned graph on (V 1 , V 2 ) in which x ∈ V 1 ∼ y ∈ V 2 ⇐⇒ ι 1 (x) = y or ι 2 (y) = x, we get a bijection f : V 1 → V 2 : this is the celebrated Cantor-Bernstein Theorem. As a proof of Cantor-Bernstein, this argument was given by Gyula ("Julius") Kőnig [Kő06] and remains to this day one of the standard proofs. His son Dénes Kőnig explicitly made the connection to matching in infinite graphs in his seminal text [Kő50]. (iii) =⇒ (iv) is the same easy argument we have already seen. (iv) =⇒ (i): By Theorem 11, we have semiderangements ι 1 : V 1 → V 2 and ι 2 : V 2 → V 1 . By Theorem 12 this gives a perfect matching.

An Equivalence.
Theorems 9 and 13 are "equivalent" in the sense that they were proved using equivalent formulations of Halls' Theorem (together with, in the case of Theorem 13, a Cantor-Bernstein argument). In this section we will show their equivalence in a stronger sense: each can readily and rapidly be deduced from the other.
Assume Theorem 9, and let G = (V 1 , V 2 , E) be a locally finite bipartitioned graph satisfying the Cockroach Condition: for all finite independent subsets J ⊂ V 1 V 2 , #N J ≥ #J. Then Theorem 9 applies to yield a surjective graph derangement f . A cycle C admits a dyadic graph derangement iff it is not finite of odd length; since G is bipartite, no cycle in f is finite of odd length. Thus decomposing f cycle by cycle yields a dyadic graph derangement.
Assume Theorem 13, and let G = (V, E) be a locally finite graph satisfying the Hall Condition: for all finite independent subsets J ⊂ V , #N J ≥ #J. Let G 2 = (V 1 , V 2 , E 2 ) be the bipartite double of G: we put V 1 = V 2 = V . For x ∈ V , let x 1 (resp. x 2 ) denote the copy of x in V 1 (resp. V 2 ). For every e = {x, y} ∈ E, we give ourselves edges {x 1 , y 2 }, {y 1 , x 2 } ∈ E 2 . Then G 2 is locally finite bipartitioned, and it is easy to see that the Hall Condition in G implies the Cockroach Condition in G 2 . By Theorem 13, G 2 admits a dyadic graph derangement f 2 . From f 2 we construct a graph derangement f of G: for x ∈ V , let x 1 be the corresponding element of V 1 ; let y 2 = f 2 (x 1 ), and let y be the element of V corresponding to y 2 . Then we put f (x) = y. It is immediate to see that f is a graph derangement of G. It need not be surjective, but no problem if it isn't: apply Proposition 2.
Remark 3.11: Our deduction of Theorem 9 from Theorem 13 is inspired by an unpublished manuscript of L. Levine [Le01].
Remark 3.12: Recall that Theorem 13 implies the Cantor-Bernstein Theorem. The above equivalence thus has the following curious consequence: Cantor-Bernstein follows almost immediately from the transversal form of Halls' Theorem! 3.5. Dyadic Existence Theorems.
One may ask if there is also an Existence Theorem for dyadic derangements in non-bipartite graphs. The answer is yes, a quite celebrated theorem of Tutte. A later result of Berge gives information on dyadic graph permutations.
Let G = (V, E) be a graph. For a subset X ⊂ V , we denote by G \ X the induced subgraph with vertex set V \ X. Proof. See [Tu47].
A maximum matching of a finite graph G is a matching M such that #M is maximized among all matchings of G. Thus if G admits a perfect matching, a matching M is a maximum matching iff it is perfect, whereas in general the size of a maximum matching measures the deviation from a perfect matching in G.
For a finite graph H, let odd(H) be the number of connected components of H with an odd number of vertices.
Theorem 15. (Berge's Theorem [Be58]) Let G be any finite graph. The size of a maximum matching in G is We immediately deduce the following result on dyadic graph permutations.
Corollary 16. Let G be a finite graph. Then the least number of fixed points in a dyadic graph permutation of G is #V − 2B G .

Matchless Graph Derangements.
Let G = (V, E) be a finite graph with #V = n. Then a graph permutation of cycle type (n) is called a Hamiltonian cycle (or Hamiltonian circuit).
From the perspective of graph derangements it is clear that Hamiltonian cycles lie at the other extreme from dyadic derangements and permutations. Here much less is known than in the dyadic case: there is no known Hamiltonian analogue of Tutte's Theorem on perfect matchings, and in place of Berge's Theorem we have the following open question.
Question 2. Given a finite graph G, determine the largest integer n such that G admits a graph permutation of type (n, 1, . . . , 1). Equivalently, determine the maximum order of a vertex subgraph of G admitting a Hamiltonian cycle.
A general graph derangement is close to being a partition of the vertex set into Hamiltonian cycles, except that if x and y are adjacent vertices, then sending x to y and y to x meets the requirements of a graph derangement but does not give a Hamiltonian subcycle because the edge from x to y is being used twice. Let us say a graph derangement is matchless if each cycle has length at least 3. The above Existence Theorems lead us naturally to the following question.
Question 3. Is there an Existence Theorem for matchless graph derangements?
As noted above, there is no known Existence Theorem for Hamiltonian cycles. Since a Hamiltonian cycle is a graph derangement of a highly restricted kind, one might hope that Question 3 is somewhat more accessible.

Checkerboards Revisited
4.1. Universal and even universal graphs.
Let G be a graph on the vertex set [n] = {1, . . . , n} such that Der G = ∅. As in §2.2 it is natural to inquire about the possible cycle types of graph derangements (and also graph permutations) of G. We say G is universal if for every partition p of n there is a graph derangement of G with cycle type n. For instance, the complete graph K n is (rather tautologously) universal.
If n ≥ 5 and G is bipartite, by Corollary 4, G is not universal, because the only possible cycle types are even. Thus for graphs known to be bipartite the more interesting condition is that every possible even partition of [n] occurs as the cycle type of a graph derangement of G: we call such graphs even universal.

4.2.
On the Even Universality of Checkerboard Graphs.
Proposition 17. For all n ∈ Z + , the checkerboard graph R 2,n is even universal.
Proof. This is an easy inductive argument which we leave to the reader.
Proposition 18. Let n ≥ 4 be an even number. Then: a) If a 1 , . . . , a k are even numbers greater than 2 such that 4 + k i=1 a i = 3n, then there is no graph derangement of R 3,n of cycle type (a 1 , . . . , a k , 4). b) It follows that R 3,n is not even universal.
Proof. a) A 4-cycle on any checkerboard graph must be a (2 × 2)-square. By symmetry we may assume that the (2 × 2)-square is placed so as to occupy portions of the top two rows of R m,n . The two vertices immediately underneath the square cannot be part of any Hamiltonian cycle in the complement of the square, so any graph derangement containing a 4-cycle must also contain a 2-cycle directly underneath the 4-cycle. Thus the cycle type (a 1 , . . . , a k , 4) is excluded. b) Since n ≥ 4 is even, 3n is even and at least 12, so there are even partitions of 3n of the above form: (4, 4, . . . , 4) if n is divisible by 4; (6, 4, 4, . . . , 4) otherwise.
Proposition 19. If m is odd and n is divisible by 4, then R m,n admits no graph derangement of cycle type (4, 4, . . . , 4) and is therefore not even universal.
Proof. Left to the reader.
Indeed, the only order 6 cycle of a checkerboard graph is the 2 × 3 checkerboard subgraph. Removing any 6-cycle leaves one of the corner vertices pendant and hence not part of any cycle of order greater than 2.
In particular, for R m,n to be even universal, it is necessary that m and n both be even. Proposition 21 shows that having m and n both even is not sufficient; however, we have found no examples of excluded even cycle types of R m,n when m, n are both even and at least 6. Such considerations, and others, lead to the following conjecture, made by J. Laison in collaboration with the author.