Uniquely $D$-colourable digraphs with large girth II: simplification via generalization

We prove that for every digraph $D$ and every choice of positive integers $k$, $\ell$ there exists a digraph $D^*$ with girth at least $\ell$ together with a surjective acyclic homomorphism $\psi\colon D^*\to D$ such that: (i) for every digraph $C$ of order at most $k$, there exists an acyclic homomorphism $D^*\to C$ if and only if there exists an acyclic homomorphism $D\to C$; and (ii) for every $D$-pointed digraph $C$ of order at most $k$ and every acyclic homomorphism $\varphi\colon D^*\to C$ there exists a unique acyclic homomorphism $f\colon D\to C$ such that $\varphi=f\circ\psi$. This implies the main results in [A. Harutyunyan et al., Uniquely $D$-colourable digraphs with large girth, Canad. J. Math., 64(6) (2012), 1310-1328; MR2994666] analogously with how the work [J. Ne\v{s}et\v{r}il and X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B, 90(1) (2004), 161-172; MR2041324] generalizes and extends [X. Zhu, Uniquely $H$-colorable graphs with large girth, J. Graph Theory, 23(1) (1996), 33-41; MR1402136].

one authoritative combinatorialist went so far as to assert that "All interesting combinatorics flows from the existence of graphs with large girth and chromatic number." 1 Of course, we interpret Thomassé's remark as somewhat tongue-in-cheek, but as they say, many a truth is said in jest.
In the present article, we follow the flow, from colourings to homomorphisms and from graphs to digraphs. This work is a sequel to [8], with which we assume some familiarity. For example, because the Introduction of [8] is more extensive than this one, we refer the reader there for more background.
Also, some of the arguments from [8]-e.g. the statement/proof of Lemma 7 and Lemma 8 (both below)-prove useful here. We try to balance the conflicting goals of not duplicating earlier work while allowing our new results to stand on their own.
Erdős' argument in [6] was probabilistic, hence nonconstructive. To help answer the question of what graphs with large girth and chromatic number actually look like, in 1968 Lovász [12] constructed hypergraphs with arbitrarily large girth and chromatic number. Müller [14] also worked in this domain. More than twenty years after Lovász's contribution, Kříž [11] produced the first purely graph-theoretic construction of graphs with arbitrarily large girth and chromatic number.
The time intervals separating these results offer some hint of the delicacy of their constructions.
Graph homomorphisms, as vertex mappings that preserve adjacency, naturally generalize graph colouring. In 1996, working in this realm, Zhu [21] proved that for every 'core' graph H and every positive integer ℓ ≥ 3 there exists a uniquely H-colourable graph with girth at least ℓ. Because complete graphs are cores, Zhu's result generalized [4] and [6]. Almost ten years later, Nešetřil and Zhu [15] further generalized the results in the sequence [6,4,21] using the notion of 'pointed' graphs.
Let us shift now to digraphs. Their circular chromatic number was first studied in [3], where Bokal et al. showed that the colouring theory for digraphs is similar to that for undirected graphs when stable vertex sets are replaced by acyclic sets. For example, using an analogue of Erdős' original argument from [6] , they showed that there exist digraphs of arbitrarily large (directed) girth and circular chromatic number. Almost a decade later, in [8], a subset of these authors together with their doctoral students established analogues of Zhu's results from [21] in a digraph setting; namely, for a suitable digraph D, there exist digraphs of arbitrarily large girth that are uniquely D-colourable. Severino [18] presented a construction of highly chromatic digraphs without short cycles and another construction [19] of uniquely n-colourable digraphs (for arbitrary n) with arbitrarily large girth. The latter two articles, based on [20], give constructive proofs of results in [3] and [8] that were originally proved probabilistically. This paper analogizes the results of Nešetřil and Zhu [15] to the realm of digraphs. Just as [15] puts the final icing on the sequence [6,4,21], so too does our main result-Theorem 1 belowprovide a fitting capstone for the sequence [3,8]. Postponing definitions for another minute (until Section 2), let us state our main result and lay bare its connection with [8]. 1 Stéphan Thomassé included the assertion in his plenary CanaDAM lecture, 2 June 2011, Victoria, Canada. Theorem 1. For every digraph D and every choice of positive integers k, ℓ there exists a digraph D * together with a surjective acyclic homomorphism ψ : D * → D with the following properties: (ii). for every digraph C with at most k vertices, there exists an acyclic homomorphism D * → C if and only if there exists an acyclic homomorphism D → C; (iii). for every D-pointed digraph C with at most k vertices and for every acyclic homomorphism ϕ : D * → C there exists a unique acyclic homomorphism f : D → C such that ϕ = f • ψ.
The precursor [8] established two main results: Theorem 2. If D and C are digraphs such that D is not C-colourable, then for every positive integer ℓ, there exists a digraph D * of girth at least ℓ that is D-colourable but not C-colourable.
Theorem 3. For every core D and every positive integer ℓ, there is a digraph D * of girth at least ℓ that is uniquely D-colourable.
To see that Theorem 1 implies Theorem 2, let us be given a positive integer ℓ and two digraphs C, D with D not C-colourable (as in the hypotheses of Theorem 2). Taking k be the order of C, we can put this C in the role of the digraph C in conclusion (ii) of Theorem 1, which delivers a digraph D * with D * → D. As D → C, the same conclusion shows that also D * → C, and conclusion (i) gives the girth requirement on D * . Before deriving Theorem 3 from Theorem 1, observe that if D is a core, then every acyclic homomorphism from D to itself must be an automorphism, and so if any two such homomorphisms agree on all but one vertex, they must also agree on that vertex. Therefore, cores D are D-pointed. Now let us be given a positive integer ℓ and a core D (as in the hypotheses of Theorem 3).
If we here take k = |V (D)|, then Theorem 1 delivers a large-girth digraph D * together with a D-colouring ψ : D * → D. The preceding paragraph foreshadows that we can put D in the role of C in conclusion (iii), which shows that every acyclic homomorphism ϕ : D * → D yields an acyclic homomorphism f : D → D such that ϕ = f • ψ. But D being a core implies that such an f is an automorphism, so we've shown that ϕ and ψ differ by an automorphism, i.e., that D * is uniquely Notice that being D-pointed is a necessary condition in part (iii) of Theorem 1. For consider two acyclic homomorphisms f ′ , f ′′ : D → C satisfying (for some vertex x 0 of D) f ′ (x) = f ′′ (x) for all x = x 0 and f ′ (x 0 ) = f ′′ (x 0 ), and assume that there is an arc between f ′ (x 0 ) and f ′′ (x 0 ) in C.
Typically, the set ψ −1 (x 0 ) can be split into two nonempty sets A, B and we can define ϕ : D * → C by f ′ • ψ(y) for y ∈ V (D * ) B and f ′′ • ψ(y) for y ∈ B. Now this ϕ sends A and B to two different points while f • ψ, for any given f : D → C, sends these sets to a single point. Therefore, the acyclic homomorphism ϕ cannot be written as ϕ = f • ψ for an acyclic homomorphism f : D → C.

Remarks
As hinted above, Nešetřil's and Zhu's article [15] was in a sense a crowning achievement for a body of work initiated by Erdős in [6]. For any given graph G, they produced a high-girth graph G * characterizing the small-order graphs admitting a homomorphism from G and furthermore, via Gpointedness, wound unique colourability into their tapestry. Their results generalized [4], [21] and moreover some other major contemporary theorems (e.g., the Sparse Incomparability Lemma and Müller's Theorem-see [21] and the discussion in [15]).
Because our Theorem 1 likewise characterizes when the high directed girth, high digraph chromatic number (for unique colourability) phenomenon occurs-phrased in terms of acyclic homomorphisms-it too reaches a satisfying destination, now for the sequence [3,8]. And because this level of generality has actually shortened the proofs from [8], perhaps we've arrived at the 'right' vantage point for viewing these results.

Terminology, notation, and an auxiliary result
Without being overly encyclopedic, we attempt to include the required definitions. For basic notation and terminology concerning graphs and digraphs, we mainly follow [5] and [2], respectively, and we refer the reader there for any omissions. For a more (most) thorough treatment of graph homomorphisms, the reader could consult [7] ( [9]). For probabilistic concerns, see, e.g., [1] or [13].
All our digraphs are finite and simple-i.e. loopless and without multiple arcs-however, we do allow two vertices u, v to be joined by two oppositely directed arcs uv, vu. Cycles in digraphs mean directed ones, and the girth of a digraph D is the length of a shortest cycle in D.
Just as graph homomorphisms generalize graph colouring, so too do acyclic homomorphisms of digraphs generalize (one variant of) digraph colouring. So we begin by recalling the definition of these sorts of homomorphisms from [3]; see [8] for background. An acyclic homomorphism of a digraph D to a digraph C is a function ρ : V (D) → V (C) such that: Acyclic homomorphisms can also be viewed as a generalization of (ordinary) homomorphisms of undirected graphs; again, see [8].
If there exists an acyclic homomorphism of D to C, we say that D is homomorphic to C and write D → C. Motivated by the connection to 'acyclic digraph colouring', we sometimes call an acyclic homomorphism of D to C a C-colouring of D and say that D is C-colourable. A digraph D is uniquely C-colourable if it is surjectively C-colourable, and for any two C-colourings ψ, ϕ of D, there is an automorphism f of C such that ϕ = f • ψ; when this occurs, we say that ϕ and ψ differ by an automorphism of C. A digraph D is a core if the only acyclic homomorphisms of D to itself are automorphisms. Given two digraphs C, D, we say that C is D-pointed if there do not exist two C-colourings ρ, ϕ of D such that ρ(v) = ϕ(v) holds for exactly one vertex v of D. As noted following the statement of Theorem 3, digraph cores D are D-pointed.

Probabilistic tools
Our proof of Theorem 1 invokes several standard probabilistic tools. Aside from the First Moment Method (Markov's Inequality)-which is explicitly invoked a handful of times-Inclusion-Exclusion and the Janson Inequalities also make an implicit appearance through their use (in [8]) in proving Lemma 8 below. We shall not restate these standard results here; however, for convenience, we do include a version of Chernoff's famous bound(s) on the tail distributions of binomial random variables. Though more technical versions are available-see, e.g., [10]-this one will suffice for our main proof in Section 4: We hope the reader will appreciate the variety of contexts for the appearance of the fifth letter of the (Greek and Roman) alphabets in this statement.
3 Set-up for the proof of Theorem 1 We begin at the starting point for the main proof in [8], namely specifying a random digraph model, which needs no change here. Suppose that the digraph D is given with V (D) = {1, 2, . . . , a} and |A(D)| = q. Let n be a positive integer and V 1 , V 2 , . . . , V a be pairwise-disjoint ordered n-sets so D 0 has na vertices and a n 2 + qn 2 arcs. Now fix an ǫ with 0 < ǫ < 1/4ℓ. Our random digraph model D(n, p) consists of all spanning subdigraphs of D 0 in which the arcs are chosen randomly and independently with probability p := n ǫ−1 . Through the following three lemmas we prove essential technical facts about digraphs in D(n, p). Throughout the discussion n is assumed to be sufficiently large to support our assertions.
Our first aim is to show that most digraphs in D(n, p) have few short cycles which are pairwise vertex-disjoint.
Lemma 5. (i). The expected number of cycles of length less than ℓ in a digraph D ∈ D(n, p) is bounded from above by n ǫℓ n −ǫ/2 ; (ii). the expected number of pairs of cycles of length less than ℓ in a digraph D ∈ D(n, p) which intersect in at least one vertex is bounded from above by n −1/2 .
By Markov's Inequality, Lemma 5 implies that asymptotically almost all digraphs from D(n, p) have at most n ǫℓ cycles of length less than ℓ, and these cycles are all vertex-disjoint. The ideas in the proofs of (i) and (ii) are contained, respectively, in the "Proof of (2.1)" and "Proof of (3.1)" in [8]; we include the proofs here for context, completeness, and consolidation.
Proof. (i) Let D ∈ D(n, p) and let the random variables X, X i count the number of cycles of length less than ℓ and length i (i ≥ 2) in D, respectively. Then recalling that p = n ǫ−1 for the last step. Now, the inequality ℓ−1 i=2 (n ǫ a) i /i < a ℓ−1 n (ℓ−1)ǫ (which can be proved by induction on ℓ) shows that E(X) < a ℓ−1 n (ℓ−1)ǫ = a ℓ−1 n −ǫ n ǫℓ < n ǫℓ n −ǫ/2 , for sufficiently large values of n.
To prove part (ii), we need the following definition from [8] which in turn had its roots in [21].
For integers ℓ 1 , ℓ 2 < ℓ, we call a digraph an (ℓ 1 , ℓ 2 )-double cycle if it consists of a directed cycle C ℓ 1 of length ℓ 1 and a directed path of length ℓ 2 joining two (not necessarily distinct) vertices of C ℓ 1 .
A moment's reflection shows that if two cycles of length less than ℓ intersect in at least one vertex, then they contain (as a subdigraph) an (ℓ 1 , ℓ 2 )-double cycle for some ℓ 1 , ℓ 2 < ℓ. Hence in a random D ∈ D(n, p) the expected number of pairs of cycles of length less than ℓ that intersect in at least one vertex is at most the expected number of all (ℓ 1 , ℓ 2 )-double cycles for ℓ 1 , ℓ 2 < ℓ.
As ǫ(ℓ 1 + ℓ 2 ) < 2ℓǫ < 1/2 (because ℓ 1 , ℓ 2 ≤ ℓ and ǫ < 1/4ℓ), for large enough n we have To state the second lemma we need the following definition (which leans on the parameters D and k of Theorem 1). This set-up and the ensuing analysis in Lemma 6 is modelled after an analogous discussion in [15]. Following these authors, we call a set A ⊆ V large if there are distinct i, j ∈ [a], with ij an arc of D, such that both |A ∩ V i | ≥ n/k and |A ∩ V j | ≥ n/k, and the D-arc ij in this case is a good arc for A. For a large set A, denote by | D/A| the minimum number of arcs of (a random) D which lie in a set {xy : for some positive constants c and c ′ that are independent of n (with the estimates in (1) being borrowed from [15]). Thus we get α = 1 − o(1).
The last lemma of this section addresses a technical situation also encountered at the end of Section 3 of [8]. We repeat part of the proof here for completeness and also to facilitate fleshing out more of its details. See also [15,Claim 3] for an analogous statement (for graphs and homomorphisms) and an alternate proof approach (via enumeration).

Lemma 7. Almost all digraphs from D(n, p) do not contain two nonempty sets
, with i 0 j 0 ∈ A(D) (resp. j 0 i 0 ∈ A(D)), |A| = n − (k − 1)|B|, |B| ≤ n/k, such that the set A ∪ B contains at most min{|B|, n ǫℓ } arcs from A to B (resp. from B to A) and these arcs form a matching (i.e. a set of independent arcs).
Proof. Let b ≤ n/k and s ≤ min{b, ⌈n ǫℓ ⌉}. We denote by L(b, s) the expected number of pairs A, and there are exactly s arcs joining a vertex in A to a vertex in B. Then To help the reader through steps (2)-(4), we fill in the following estimates: for (2): for (3): for large enough n, we have n k < e n ǫ/6 , so that n k e −n ǫ/2 < e −n ǫ /3 ; and lastly for (4)

Proof of Theorem 1
We continue to be guided by [15], but the argument here is complicated by the more technical definition of 'acyclic homomorphism' in our context compared to 'homomorphism' in the graph setting.
Choose a digraph D ′ in D(n, p) satisfying the properties asserted in Lemmas 5-7. So D ′ contains at most n ǫℓ (directed) cycles of length less than ℓ and these cycles are pairwise vertex-disjoint.
Consequently (picking one arc from each cycle), there is a matching (an independent arc set) , we have |A i | ≥ n/k and |A j | ≥ n/k from the definition of f . Hence A = A i ∪ A j is a large set and ij is a good arc for A, so we can invoke Lemma 6 to see that there exists an arc of D * from A i to A j (Note that we deleted at most n ǫℓ < n 1/4 arcs from D ′ to get D * , but ij induces at least n arcs, so we did not delete all these arcs from A i to A j ). Now, since ϕ is an acyclic homomorphism, we have xy ∈ A(C) as required.
To finish the proof that f is an acyclic homomorphism, we need to show that f −1 (x) induces an acyclic subdigraph in D for every x ∈ V (C). We prove this by contradiction. Suppose that there is a vertex v ′ ∈ V (C) such that the subdigraph induced by f −1 (v ′ ) in D contains a cycle Q.
Write Q = i 1 i 2 · · · i t and observe that 2 ≤ t ≤ a. Since f (i s ) = v ′ , for s = 1, 2, . . . , t, we have |V is ∩ ϕ −1 (v ′ )| ≥ n/k, for s = 1, 2, . . . , t (from the definition of f ). The fact that n ǫℓ ≪ n/k implies that each set V is ∩ ϕ −1 (v ′ ) contains a subset W is of size w := ⌈n/(2k)⌉ such that no arc in M has ϕ is an acyclic homomorphism, the subdigraph of D * induced by W i 1 ∪ W i 2 ∪ · · · ∪ W is is acyclic.
We show that the event that W i 1 ∪ W i 2 ∪ · · · ∪ W is induces an acyclic subdigraph in D * is unlikely.
Let us consider a sequence of sets U j 1 , U j 2 , . . . , U jr such that for i = 1, 2, . . . , r we have U j i ⊆ V j i and |U j i | = w, and the vertex sequence j 1 , j 2 , . . . , j r is a cycle in D. We denote by P r the probability that the subdigraph of D * induced by U j 1 ∪ U j 2 ∪ · · · ∪ U jr is acyclic and call this sequence bad if it induces an acyclic subdigraph in D * . Now, for the expected number N of bad sequences in D * , we We pause to note that (5) is relation (2.6) from [8], adapted to our present context. The following result bounds the probabilities P r ; for a proof, see [8] (which actually contains two proofs).
As in [8, relations (2.19)], we see that for large enough n, the relation (5) and Lemma 8 give So to finish this chain of reasoning as in [8], using Markov's Inequality, we find that P (∃ a bad sequence) < e −n 2 .
This achieves the goal stated before Lemma 8 which in turn contradicts our assumption that the subdigraph induced by f −1 (v ′ ) in D contains a cycle. Thus the forward implication in part (ii) of Theorem 1 is proved.
For the converse in (ii), let f : V (D) → V (C) be an acyclic homomorphism. We define a mapping Remark Until now, parts of our proof have involved carefully piecing together ideas from [8] and [15]. The remainder of the argument follows quite a different path and underscores the extra complexity inherent in working with acyclic homomorphisms (of digraphs) compared to ordinary homomorphisms (of graphs).
Proof of ϕ = f • ψ First, we show that ϕ and f • ψ have the same range. It is clear that Range(f • ψ) ⊆ Range(ϕ).
To prove the reverse containment, suppose to the contrary that there is a vertex y ∈ Range(ϕ) that is not in Range(f • ψ). Since y is in the range of ϕ, the set ϕ −1 (y) ∩ V i is not empty for some i ∈ {1, 2, . . . , a}. On the other hand, the definition of f shows that |ϕ −1 (f (i)) ∩ V i | ≥ n/k; in . . , a}. We show that this leads to a contradiction.
Let i 0 ∈ {1, 2, . . . , a} be such that t := |ϕ −1 (f (i 0 )) ∩ V i 0 | is minimum. It is easy to see that t ≥ n/k. Our discussion in the preceding paragraph implies that t < n. We choose x ∈ V (C) with Using the Pigeonhole Principle we obtain b ≥ (n − t)/(k − 1) which gives t ≥ n − (k − 1)b. Furthermore b < n/k as there is only one vertex of V (C) satisfying the negation (f (i 0 ) = x already has this property). Now we define a mapping Since f and f ′ differ only at i 0 and C is D-pointed, the function f ′ cannot be an acyclic homomorphism. We distinguish two cases.
Case I: x ∈ Range(f ). In this case, the only reason that f ′ is not an acyclic homomorphism is that there must be a there must be at most min{b, n ǫℓ } arcs from U to A := ϕ −1 (x) ∩ V i 0 in D ′ ; otherwise after passing from D ′ to D * , we have some arc(s) left between these two sets in D * and since ϕ is an acyclic homomorphism, f (v)x ∈ A(C) which is a contradiction. But the property just described is the rare property articulated in Lemma 7, and D ′ was chosen not to enjoy it, so Case I leads to this contradiction.
Case II: x ∈ Range(f ). We know that ϕ −1 (x) ∩ V i 0 = ∅. Since x ∈ Range(f ), we have x = f (j 0 ) for some j 0 ∈ V (D) The definition of f gives us To show that this is unlikely, we first estimate the expected number N of ways to select a vertex z ∈ V i 0 and a subset U ⊆ V j 0 of cardinality ⌊n/2k⌋ so that the subdigraph H z,U of D * they induce is acyclic and no arc of M is incident with a vertex in U. If P z,U denotes the probability that H z,U is acyclic, then N ≤ n n ⌊n/2k⌋ P z,U < n n P z,U .
In order to bound P z,U , we employ Chernoff's Inequality (Theorem 4). Let Ω be the set of all potential arcs in the subdigraph D ′ z,U , of D 0 induced by {z} ∪ U. Each arc in Ω appears in H z,U with probability p. Let τ > (2 + ǫ)/ǫ be a fixed integer. We index (by positive integers) those cycles of D ′ z,U that are of length τ + 1. For i ≥ 1, let S i be the arc set of the ith such cycle and B i be the event that the arcs in S i all appear (i.e., the cycle determined by S i is present in H z,U ). Let the random variable Y count the B i 's that occur. Since P (Y = 0) is an upper bound for P z,U , we can bound P z,U by bounding P (Y = 0). Using Theorem 4 with ǫ = 1, we have Since the arcs of D ′ z,U within U are acyclically oriented, each choice of τ vertices within U determines exactly one potential (τ + 1)-cycle. It follows that Using (7) and (8) we find that P (Y = 0) ≤ 2e −n ǫτ +ǫ−1 /3(4kτ ) τ , and recalling our choice of τ (as exceeding (2 + ǫ)/ǫ), we see that P (Y = 0) ≤ 2e −n 1+2ǫ /3(4kτ ) τ < e −n 1+ǫ .
By Markov's Inequality, the probability that there exists such a set {z} ∪ U that induces an acyclic subdigraph is less than e −n 1+ǫ /2 , which means it is unlikely as desired.
Our discussion in Cases I and II implies that ϕ and f • ψ have the same range. It is now evident that ϕ = f • ψ, for otherwise the same situation as in the proof that Range(ϕ) = Range(f • ψ) occurs and similarly leads to a contradiction. Hence ϕ = f • ψ as desired and therefore the proof of part (iii) of Theorem 1 is complete. ✷