MARK H. SIGGERS
Abstract. In this note, we present a new proof of theH-colouring Dichotomy, which was proved by Hell and Neˇsetˇril in 1990, and was reproved by Bulatov in 2005. The proof is much shorter than the original proof, and avoids the algebraic machinery of Bulatov’s proof.
1. Introduction
Arguably, one of the most celebrated results in graph homomorphism theory is the H-colouring Dichotomy, or Graph Homomorphism Dichotomy, of Hell and Neˇsetˇril, [3].
A homomorphism of a graph G to a graph H, or an H-colouring of G, is a mappingφof the vertices ofGto the vertices ofH such thatφ(u)φ(v) is an edge of H ifuvis an edge ofG. The problemH-COL is a computational problem in which one is given an instance, a graphG, and one must determine if there exists anH- colouring ofG. The question of interest for a givenH, is what is the computational complexity of the problemH-COL, in terms of the size ofG.
The following result of Hell and Neˇsetˇril, stated here for loopless graphs, is known as theH-colouring Dichotomy.
Theorem 1.1. [3]The problem H-COL is polynomial time solvable ifH is bipar- tite, and isN P-complete otherwise.
The proof of Hell and Neˇsetˇril uses purely graph theoretic techniques, but is fairly long and very intricate. In [1], Bulatov used techniques from Universal Algebra, popularised in [2, 5], to simplify the proof of Hell and Neˇsetˇril. His reproof was a byproduct of showing that the H-colouring Dichotomy agreed with the more general CSP Dichotomy Classification conjectured in [2].
In [9, 10], a construction called the ’fibre construction’ was developed, and was used to give low machinery proof of many relevant results previously proved by alge- braic means. In particular, a graph theoretic interpretation of the CSP Dichotomy Classification Conjecture was given.
In this note, we reprove theH-colouring Dichotomy using a variation of the fibre construction from [10]; or more accurately, we prove Proposition 3.1, which replaces the hardest part of the original proof.
This is a clarification of the published version, which appears in SIAM J. Discrete Math.
Volume 23, Issue 4, pp. 2204-2210 (2010), available at http://dx.doi.org/10.1137/080736697. The published version has several typos, (which I introduced after the paper was refereed,) mostly concerning the notation p1(G) (and p2(G)) which is defined as one more than the number of vertex (resp. vertex pair) orbits ofG, but is often treated as the number of orbits. In this version, I have corrected these mistakes, and further, have seperated results unnecessary to the proof of the dichotomy, from the main part of the paper.
Our new proof retains important ideas introduced in both of the earlier proofs, but is shorter than the proof of [3] and does not require the algebraic machinery used in [1]. As well as being a streamlined graph theoretic proof of Theorem 1.1, our proof gives a solid intuition as to why certain reductions are natural. Furthermore, we view this proof as further evidence that the fibre construction of [10] will be a useful tool in the study of CSPs.
2. Notation and definitions
All graphs are simple, loopless and undirected. We denote an edge containing the vertices uand v by uv, and denote the fact that it is an edge by u∼v. An H-colouringφof a graphGis an injection ifφ(u) =φ(v) implies thatu=v. An automorphism of H is an injection fromH toH. We writeG→H if there exists an H-colouring of G. We denote the fact that S is a subgraph of G by S ≤ G.
A graph C is a coreif its onlyC-colourings are automorphisms. It is well known (see [4]), and easy to show, that every graphH has a unique (up to isomorphism) subgraphHCwhich is a core. FurthermoreH →HCandHC→H, soH-COL has the same computational complexity asHC-COL.
The complete graphK3 will always have vertex set {1,2,3}. When we denote a functionf acting on a vector (v1, . . . , vd) we drop a set of parentheses and write f(v1, . . . , vd).
3. From Theorem 1.1 to Proposition 3.1
AsK2 (orK1) is the core of every bipartite graph, it is easy to see thatH-COL is polynomial time solvable for bipartite graphs. Indeed, K2-COL is equivalent to checking if the instance Gis bipartite, and K1-COL is equivalent to checking for the existence of edges.
The hard part of Theorem 1.1 is showing that H-COL is N P-complete if H contains an odd cycle. Hell and Neˇsetˇril’s first step in proving this was reducing it to the case in whichH has the following properties:
• H is a core.
• Every vertex inH is in a triangle.
• H contains no copies ofK4.
These reductions are standard, and Bulatov follows the proof exactly up to this point, but here the proofs diverge. However, with different proofs, they both go on to further reduce to the following case:
• Every edge ofH lies in a unique triangle.
The details of these reductions can be found in any number of places, among others, [3, 4] and [1], and we add nothing to them. We thus pick up the proof from here, that is, we prove the following.
Proposition 3.1. H-COL isN P-complete for any core graph H, with at least one edge, in which every edge lies in a unique triangle.
The proof of Proposition is immediate from Lemmas 4.2 and 5.1, which we prove in Sections 4 and 5 respectively. Section 6 contains a generalisation of the ideas in Section 4.
4. A Lemma about Projectivity
Definition 4.1. For an integer d≥1 thedth powerK3d of K3 is the graph with vertex set
{(x1, . . . , xd)|x1, . . . xd∈ {1,2,3}},
and in which (x1, . . . , xd)∼(x01, . . . , x02) if and only ifxi6=x0i fori= 1, . . . , d.
A homomorphismφ : K3d → H is called projective if it projects onto some for some non-emptyindex setI⊆[d]; that is, if
φ(x1, . . . , xd) =φ(x01, . . . , x0d)
if and only ifxi=x0ifor alli∈I. In particular, ifφis an injective homomorphism, it is a projection.
By the symmetry ofK3d, it is clear that any mapσ:K3d→K3d which permutes coordinates, or permutes the entries in any coordinate, or is the composition of such operations, is an automorphism ofK3d. Further, it is clear that an H-colouringφ ofK3d is projective if and only ifφ◦σis for any such automorphismσ.
The following lemma is implicit in Bulatov’s reproof of the H-colouring Di- chotomy. However, where his proof uses the algebraic machinery developed in [2], we give a simple proof. By Lemma 6.4 it is enough to prove in the case thatd= 2,3, and this is what we did in the published version of the paper. However, it is clearer to prove it directly here.
Lemma 4.2. Let H be a graph in which no edge occurs in more than one triangle.
Then allH-colourings of K3d are projective, for alld≥2.
Proof. We first prove the cased= 2, and then proceed by induction ond.
Case: d= 2. Letφ be an H-colouring of K32. If φis an injection, we are done, so by symmetry, we may assume thatφmaps (1,1) and (1,2) to the same image.
The following figure showsK32 in which these vertices have been identified.
(2,2) (3,3) (2,1)
(1,3) (1,1) = (1,2)
(3,1) (2,3) (3,2)
Observe that any two vertices that have the same entry in the first coordinate are non-adjacent vertices occurring in distinct triangles with a common edge, thus must be mapped to the same image inH. Identifying such pairs of vertices, we find thatφis a projection onto the first coordinate.
Now assume that d ≥3 and that any homomorphism K3c → H 2 ≤ c < d is projective. Letφ be anH-colouring of K3d. If φ is injective then we are done, so up to symmetry we may assume that φ(1,1, . . . ,1) = φ(1, . . .1,2, . . . ,2) where in
the second vertex, there arerones. We will show thatφis independent of the last coordinate; that is, that for any choice of entries,
φ(x1, . . . , xd) =φ(x1, . . . , xd−1, x0d) (1) Case: r >1. First we show that (1) holds whenx1=x2. Letφ0(x2, x3, . . . , xd) = φ(x2, x2, . . . , xd). Asφ0 is projective by induction, and
φ0(1, . . . ,1) =φ(1, . . . ,1) =φ(1, . . . ,1
| {z }
r
,2, . . . ,2) =φ0(1, . . . ,1
| {z }
r−1
,2, . . . ,2), φ0 projects onto an index set not containing the last index. Thus
φ(x2, x2, . . . , xd) =φ0(x2, . . . , xd) =φ0(x2, . . . , xd−1, x0d) =φ(x2, x2, . . . , xd−1, x0d) and so (1) holds whenx1=x2. We now show that also (1) holds whenx16=x2. For anyiletxi, x0i,andx00i be distinct vertices ofK3. So we may writex1=x002. Then a = (x002, x2, x3, . . . , xd), a0 = (x02, x02, x03, . . . , x0d) and a00 = (x2, x002, x003, . . . , x00d) are a triangle, anda00,b= (x002, x2, x3, . . . , xd−1, x0d), andb0= (x02, x02, x03, . . . , x0d−1, x00d) are a path. The same is true after applying φ, but φ(a0) = φ(b0), so have the following graph in the image ofφ.
φ(a0) =φ(b0)
φ(a) φ(b)
φ(a00)
But this graph doesn’t exist inH, soφ(a) =φ(b), which is (1) (becausex1=x002).
Case: r= 1. The proof is similar to the caser >1. Usingφ0(x1, x2, x4, . . . , xd) = φ(x1, x2, x2, . . . , xd) we get that (1) holds in the case thatx2=x3. In the case that x26=x3=x002, we have the following graph in the image ofφ:
φ(x01, x01, x02, x04, . . . , x0d) =φ(x01, x01, x02, x04, . . . , x0d−1, x0d)
φ(x1, x2, x002, x4, . . . , xd) φ(x1, x2, x002, x4, . . . , xd−1, x0d)
φ(x001, x002, x2, x004. . . , x00d)
In the same way as in the previous case, this shows that (1) holds.
As (1) holds,φ(x1, . . . , xd) =φ0(x1, . . . , xd−1) for someH-colouringφ0 ofK3d−1, which is projective by induction. So φ is projective. This completes the proof of
the lemma.
5. The Fibre Construction
We prove the following lemma using a version of the Fibre Construction from [10], which we have tailored to the present problem.
Lemma 5.1. IfH is a core containing a copy ofK3 such that anyH-colouring of K3d is projective for alld≥2, thenH-COL is N P-complete.
Proof. We present a polynomial time construction which provides, for an instance GofK3-COL, an instanceM(G) ofH-COL, such thatG→K3 ⇐⇒ M(G)→H. AsK3-COL isN P-complete [6], this will show thatH-COL isN P-complete.
Letdbe the maximum integer such thatH contains a copy ofK3d, and letT be a fixed copy of K3d inH. Let the vertices of T bed-tuples ˙x= (x1, . . . , xd) of the set{1,2,3}, such that ˙x∼y˙ if and only if all coordinates are different.
In the construction, copies of the graph T ∼= (K3d)6, will figure prominently.
Vertices ofT are 6-tuples ofd-tuples of the set{1,2,3}. We give special labels to the following two vertices:
¨
a= ((1, . . . ,1),(1, . . . ,1),(2, . . . ,2),(2, . . . ,2),(3, . . . ,3),(3, . . . ,3)),
¨b= ((2, . . . ,2),(3, . . . ,3),(1, . . . ,1),(3, . . . ,3),(1, . . . ,1),(2, . . . ,2)).
Given a copy Te of T, the copy of a vertex ( ˙x1, . . . ,x˙6) is labelled ( ˙x1, . . . ,x˙6)e and in particular, the copies of ¨aand ¨b are labelled ¨aeand ¨be.
Construction 5.2. Given a graphG, arbitrarily orient its edges. For every arce ofGletTe be a copy ofT. BuildM(G)from G,H andTefor eache∈E(G), as follows.
(i) For each arc e=uv∈E(G), identifyuandv ofGwith ¨aeand¨berespec- tively.
(ii) For each e∈ E(G) and each vertex x˙ ∈V(T), identify ( ˙x, . . . ,x)˙ e in Te
with x˙ inT ≤H ≤M(G).
We show that G→K3 ⇐⇒ M(G)→H. Letφ be anH-colouring of M(G).
AsT ∼=K3d isK3-colourable, the following claim implies thatG isK3-colourable, giving us the implicationM(G)→H ⇒G→K3.
Claim 5.3. The homomorphismφmapsG≤M(G)to a subgraph ofH isomorphic toT.
Proof of claim. Let uv be an arc of G. Since φ maps H ≤ M(G) to H and H is a core, we may assume that φ is the identity on H. By the identification of ( ˙x, . . . ,x)˙ uv ∈ Tuv with ˙x ∈ T ≤ H ≤ M(G) we have that V(T) ⊆ φ(Tuv).
Moreover sinceTuv is isomorphic toK36d, so by the premise of the lemma maps to a power of K3 in H, and T is contained in no greater power ofK3 in H,φ maps Tuv ontoT. In particularφ maps the edge ¨auv¨buv ofTuv to an edge of T. Butu and v are identified with ¨auv and ¨buv respectively, so φ mapsuv to an edge ofT.
To see the implication G→K3 ⇒M(G)→H, we let φ be aK3-colouring of G, and define an H-colouring φ0 of M(G) as follows. Let φ0 be the identity on H ≤M(G). Extendφ0 toG≤M(G) by lettingφ0(v), forv∈V(G), be thed-tuple (φ(v), . . . , φ(v)) inT ≤H. We have just to verify that φ0 can be extended to an H-colouring ofTe for each e =uv ∈E(G). The only vertices ofTe on which φ0
has already been defined, are ¨ae and ¨be, and ( ˙x, . . . ,x)˙ e for each ˙x∈ T. Indeed, φ0(¨ae) = (φ(u), . . . , φ(u)) andφ0(¨be) = (φ(v), . . . , φ(v)) ared-tuples inT, whereφ(v) andφ(u) are different elements ofi∈ {1,2,3}, andφ0(( ˙x, . . . ,x)˙ e) = ˙x∈T ≤H for
˙
x∈T. Recalling the definition of the vectors ¨aand ¨b, one sees that these mappings are consistent with a projection ofTeonto one of its six coordinates. We may thus extendφ0 to aT-colouring ofTeby extending it to this projection. Doing this for allTewe extendφ0 to anH-colouring ofM(G), as needed.
6. More Projectivity
The proof of the dichotomy is complete. This section expands on the ideas from Section 4, and is of independent interest.
Definition 6.1. Given a graphH and an integerd≥1 thedth(categorical) power Hd of H is defined as the graph whose vertex set is the set V(H)d of d-element vectors overV(H), and whose edges are defined by
(u1, . . . , ud)∼(v1, . . . , vd) ⇐⇒ ui∼vi for alli= 1, . . . , d.
AnH-colouringφofHd is called a (d-ary)polymorphismofH.
It is easy to check that for any i ≤ d, the projection πi of Hd defined for all vectors ˙v = (v1, . . . , vd) ∈ V(Hd) by πi( ˙v) = vi, is a polymorphism. A graph H is called projective if its only polymorphisms are projections composed with automorphisms ofH. In [5], Jeavons showed that ifH is a projective graph ( any projective relational structure, in fact), thenH-COL isN P-complete.
In [8] it was shown for any connected graphGwith at least 3 vertices, that if all 2-ary polymorphisms ofGare projections, then all polymorphisms are projections.
We now look at the same concept, but for homomorphisms of powers of Gto an arbitrary targetH, rather than just toG.
Definition 6.2. For anyI ={α1, . . . , α|I|} ⊆ {1,2, . . . , d} with αi < αi+1 for all i, define the(generalised) projectionπI :Hd→H|I|, by letting
πI( ˙v) = (vα1, . . . , vα|I|) for all vectors ˙v= (v1, . . . , vd)∈V(Hd).
It is again easy to check that this is a homomorphism.
Definition 6.3. AnH-colouringφofGd is calledprojectiveifφ=ı◦πI for some projectionπI :Gd→G|I|, with|I| ≤d, and some injectionı:G|I|→H.
Given a graphG, letp1(G) denote the number of orbits of vertices, under the group of automorphisms ofGand letp2(G) denote the number of orbits of ordered pairs of vertices. Thus p1(Kk) = 1, p2(Kk) = 2 and for any G, p21(G)≤p2(G)≤
|V(G)|2.
Lemma 6.4. For any graphsGandHif allH-colourings ofGp2(G)+1are projective then all H-colourings of any power ofGare projective.
Proof. We show by induction onnthat anyH-colouringφofGn is projective. All H-colourings ofGp2(G)+1being projective clearly implies allH-colourings ofGdare projective ford < p2(G) + 1, so we may assume thatn≥p2(G) + 2.
Ifφis independent of any coordinate, then it factors through an H-colouring of Gn−1, so is projective by induction. Thus we may assume that it depends on every coordinate.
Claim 6.5. For any choice of1< i < j≤n, and any automorphismτ of G, there exist vertices v1, . . . , vn and v10 6=v1, such that vj =τ(vi) and φ(v1, v2, . . . , vn)6=
φ(v10, v2, . . . , vn).
Proof of claim. Asφdepends on the first coordinate, there are verticesu1, . . . , un and u01 6=u1 such that φ(u1, u2, . . . , un)6=φ(u01, u2, . . . , un). As n−2≥p2(G)>
p1(G) there are some 1 < s < t ≤ n, s 6= i such that ut = σ(us) for some automorphismσofG. Define anH-colouringφ0 ofGn−1by
φ0(x1, . . . , xn−1) =φ(x1, . . . , xt−1, σ(xs), xt. . . , xn−1).
By induction,φ0 is projective, and because
φ0(u1, u2, . . . , ut−1, ut+1, . . . , un) = φ(u1, u2, . . . , un) 6= φ(u01, u2, . . . , un)
= φ0(u01, u2, . . . , ut−1, ut+1, . . . , un), it projects onto some index setIcontaining 1. Thusφ(x1, . . . , xn)6=φ(x01, . . . , x0n) whenever xt = σ(xs), x0t = σ(x0s), and x1 6= x01. In particular φ(v1, . . . , vn) 6=
φ(v10, . . . , vn) for any choice of v1, . . . , vn and v10 6= v1 such that vj = τ(vi) and vt=σ(vs). Asi6=s, there is some such choice.
Claim 6.6. Wheneverx16=x01,φ(x1, . . . , xn)6=φ(x01, . . . , x0n).
Proof of claim. Becausen−1≥p2(G), there exist some 1< i < j≤n, and some automorphismσofG, such thatxj=σ(xi) andx0j=σ(x0i). By the previous claim, there exist v1, . . . , vn and v10 6= v1, such that vj = σ(vi) and φ(v1, v2, . . . , vn) 6=
φ(v10, v2, . . . , vn). Using this fact, we can show, just like in the previous claim, that theH-colouringφ00 ofGn−1 defined
φ00(x1, . . . , xn−1) =φ(x1, . . . , xj−1, σ(xi), xj. . . , xn−1),
projects onto some index set I containing 1. Thus φ(x1, . . . , xn)6=φ(x01, . . . , x0n).
By symmetry, claims similar to Claim 6.6 are true for any index in place of 1.
Thusφis an injection, and so projective.
This upper bound on the degree we have to check to decide if there are non- projective homomorphisms from powers ofG to H is not tight. In particular, for any clique Kk we have thatp2(Kk) + 1 = 3, but in Lemma 4.2 we showed that if all H-colourings of Kk2 are projective, then all H-colourings ofKkd are projective for anyd.
It would be interesting to find exactly what degree one has to check for a given graphG.
7. Remarks and Acknowledgments
•Lemma 5.1 could be proved for all graphsP in place ofK3, and could be proved more quickly by showing that H is K3-partitionable, as defined in [10]. However we prove it directly to avoid the heavier definitions of this paper. See below for details.
• A new, completely different proof of the H-colouring Dichotomy is given in a recent paper of Kun and Szegedy [7].
•I would like to express my gratitude to G´abor Tardos for many suggestions which improved the presentation. More importantly, I thank him for his help in improv- ing the proof of Lemma 6.4 and for his observation that it can be proved for all graphs, rather than just cliques. I would also like to thank Pavol Hell for urging me to publish this, and him, Andrei Bulatov, and Jarik Neˇsetˇril for offering useful comments.
•I would like to thank the referee’s of the original published paper for their helpful comments.
8. Added 2014
I have been asked about the comment that Lemma 5.1 could be proved more quickly by showing thatH isK3-partitionable. Here are some details.
In [10] we give the following definition.
Definition 8.1. A structure H as K3-partitionableif there is
• an instance M ofH-COL containing disjoint ordered setsWa and Wb of dvertices each,
• disjoint familiesP1,P2,P3of orderedd-tuples ofV(H) (so-calledpatterns),
• such that any homomorphism φ : M → H restricts on Wa and Wb to patterns in distinct familiesPi andPj, and
• for any choice of distincti, j∈ {1,2,3}there is a homomorphismφ:M → H that retricts onWa to a pattern inPi and onWb to a pattern in Pj. In [10] we show that ifH has ak3-partition, then one can reduceK3-COL toH- COL, showing the latter is N P-complete. Lemma 5.1 could be proved by showing thatH isK3-partitionable, as follows.
Proof. Wheredis the greatest integer such thatH contains a copy ofK3d, andT is a fixed copy ofK3d inH,Gbe constructed from the graphT = (K3d)6 and a copy ofH0by identifying the diagonal copy ofT inT (the one on the vertices ( ˙x, . . . ,x)˙ ) withT ≤H0.
We show thatGis aK3-partition ofH. The singletons{¨a}and{¨b}are setsWa andWb respectively, and Pi is the family ofd-tuples whose first element isi.
Any homomorphismφ:G→H maps the coreH0 identically (we may assume,) to H, so maps the diagonal copy of T in T to T ≤H. The rest of T = (K3d)6 must map projectively onto T, so φ restricted to T is a projection onto d slots.
Identification of the diagonal T in T with T in H ensures that these slots must have indicesi1, . . . i6whereijis some number congruent tojmodulo 6. Whichever the first of these slots is,φmaps ¨aand ¨bto patterns in different familiesPiandPj.
That such a map exists for any choice ofi andj is trivial.
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Accepted to European Journal of Combinatorics (2009).
E-mail address:[email protected]
Kyungpook National University Department of Mathematics College of Natural Sciences Daegu 702-701, Republic of Korea
