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11
Hypergraph colouring
CSILLA BUJT´
AS, ZSOLT TUZA and VITALY VOLOSHIN
1. Introduction
2. Proper vertex- and edge-colourings
3. C-colourings
4. Colourings of mixed hypergraphs
5. Colour-bounded and stably bounded hypergraphs
6. Conclusion
References
We discuss the colouring theory of finite set systems. This is not merely
an extension of results from collections of 2-element sets (graphs)to larger
sets. The wider structure (hypergraphs)offers many interesting new kinds of
problems, which either have no analogues in graph theory or become trivial
when we restrict them to graphs.
1. Introduction
In this introductory section we give the most important definitions required
to study hypergraph colouring, and briefly survey the half-century history of
this topic. For more details on the material of Sections 1 and 2 we refer to
Berge [8], Zykov [76] and Duchet [27].
Let V={v1, v2,...,vn}be a finite set of elements called vertices, and let
E={E1, E2,...,Em}be a family of subsets of Vcalled edges or hyperedges.
The pair H= (V, E) is called a hypergraph with vertex-set V=V(H) and
edge-set E=E(H). The hypergraph H= (V, E) is sometimes called a set
system. If each edge of a hypergraph contains precisely two vertices, then it
is a graph. As in graph theory, the number |V|=nis called the order of the
hypergraph. Edges with fewer than two elements are usually allowed, but
will be disregarded here. Thus, throughout this chapter we assume that each
edge E∈ E contains at least two vertices, unless stated explicitly otherwise.
Edges that coincide are called multiple edges.
1
In a hypergraph, two vertices are said to be adjacent if there is an edge
containing both of these vertices. The adjacent vertices are sometimes called
neighbours of each other, and the set of neighbours of a given vertex vis
called the (open) neighbourhood N(v) of v. If v∈E, then the vertex vand
the edge Eare incident with each other. For an edge E, the number |E|is
called the size or cardinality of E.
If every edge of His of size r, then His called an r-uniform hypergraph;
evidently, a simple graph is a 2-uniform hypergraph. For 2 ≤r≤n, we
define the complete r-uniform hypergraph to be the hypergraph Kr
n= (V, E)
for which |V|=nand Eis the family of all subsets of Vof size r. Thus, the
complete 2-uniform hypergraph K2
nis the complete graph Kn.
Traditionally, for a set Xand an integer k≥1, the family of all k-element
subsets of Xis denoted by X
k. With this notation we have Kr
n= (V, V
r).
A subset of vertices S⊆Vis called stable (or independent) if no edge of
His a subset of S. The maximum cardinality of a stable set is termed the
stability number (or independence number)α(H) of H.
Aproper λ-colouring of a hypergraph H= (V, E) is a mapping c:V→
{1,2,...,λ}for which every edge E∈ E has at least two vertices of different
colours. The number of proper λ-colourings of His a polynomial in λ; it is
denoted by P(H, λ) and is called the chromatic polynomial. The minimum
value of λfor which there exists a proper λ-colouring of a hypergraph His
called the chromatic number of H, denoted by χ(H). A hypergraph His
k-colourable if χ(H)≤k, and is k-chromatic if χ(H) = k. When χ(H)≤2,
the hypergraph His called bicolourable. (In parts of the literature the term
‘bipartite’ is also used.)
As for graphs, proper colourings generate partitions of the vertex-set into
a number of stable (monochromatic) non-empty subsets called colour classes,
with as many classes as the number of colours actually used.
A graph G= (V, E) is called a host graph of a hypergraph H= (V, E)
if each edge of Hinduces a connected subgraph in G. Some structural sub-
classes of hypergraphs are identified referring to this notion. In particular,
interval hypergraphs, hyperstars, hypertrees and circular hypergraphs are de-
fined as hypergraphs with a host graph that is a path, star, tree or cycle.
Since 1-element edges are excluded, a ‘hyperstar’ means that the intersection
of all edges is non-empty.
Historical background
2
The above definitions represent natural generalizations of the respective con-
cepts from graph theory. It was realized in the early 1960s that many graph-
theoretical methods can be successfully extended to the more general struc-
ture of set systems. The term ‘hypergraph’ was first suggested at the seminar
series of Claude Berge [9].
The colouring of hypergraphs started in 1966 when Erd˝os and Hajnal [31]
introduced the notions of colouring and the chromatic number of a hyper-
graph, and obtained the first important results. In particular, they defined
the colouring number, whose graph analogue is also called the ‘Szekeres–Wilf
number’ and is equal to the ‘degeneracy’ + 1; this became an important tool
in studying the chromatic number of a hypergraph. From the definition, in
a proper colouring no edge is allowed to be monochromatic. In the literature
these colourings are sometimes called weak colourings.
This generalization of graph colourings initiated a wide area of further re-
search. First, some old problems in set systems were formulated as colouring
problems and many results in graph colouring were extended to hypergraphs.
For example, using a natural generalization of the degree of a vertex, the clas-
sic theorem of Brooks (see Chapter 2) was shown to hold for hypergraphs (see
[8], [76]). Another generalization of the degree of a vertex allowed Tomescu
[61] to extend to hypergraphs some advanced upper bounds on the chromatic
number from graph results of Welsh and Powell [71]. Berge [8] introduced
the β-degree of a vertex and showed that by using it, many previous bounds
follow, including those related to the chromatic number. In addition, it
provided an algorithm for finding a bound on the chromatic number of a
hypergraph.
It is well known that bipartite graphs play a significant role in graph
theory. A simple but very important characterization of them is the one by
K¨onig, that a graph is bipartite if and only if it contains no odd cycles. For
hypergraphs, there is no such simple condition for bicolourability, although
some criteria in terms of transversal hypergraphs were found by Berge [8]. In
the next section we present some further important sufficient conditions on
bicolourability. Also, the bicolourable hypergraphs were the first application
of the widely used Lov´asz local lemma [32]. Many more classes of hypergraphs
generalizing bipartite graphs have been investigated; see [8] for details.
It is generally accepted that graph colouring started in 1852 with the
question of whether any map can be coloured using just four colours (see
Chapter 1). It eventually led to the concept of a planar graph. In 1974 Zykov
[76] introduced the notion of a planar hypergraph and obtained the first
3
results on them. He also generalized the notions of connection–contraction
and of the Hadwiger number of a graph (see Chapter 4). The latter makes it
possible to consider hypergraph versions of problems related to Hadwiger’s
conjecture.
Sometimes a hypergraph approach permits problems to be solved more
easily than a graph approach. A celebrated example is Berge’s weak perfect
graph conjecture, which was first proved by Lov´asz [46] using so-called normal
hypergraphs.
Many generalizations of graph colourings to hypergraphs are more spe-
cific and more restrictive than the Erd˝os–Hajnal approach. We discuss a
few of these in the following sections. However, there is one thing that they
have in common: they all have graph analogues, because they all emerged
from a single concept – graph colourings: this means that they all exclude
monochromatic edges. As we see next, a new approach that is even more
general than the Erd˝os–Hajnal generalization allows us to introduce colour-
ings that have no analogues for graph colouring. In part, this is because, by
assumption, the possibility of monochromatic edges is not excluded.
We may observe further that the Erd˝os–Hajnal classic concept of hyper-
graph colouring is asymmetric: with the chromatic number as its central
notion, this theory focuses on the minimum number of colours, while the
maximum number of colours has no mathematical interest since a totally
multicoloured vertex-set is always feasible. This asymmetry of classical hy-
pergraph colouring was inherited from graph colouring.
A different direction for research concerns the maximum number of colours,
excluding polychromatic edges. In hypergraph terms this was proposed by
Berge and was first investigated in detail by Sterboul [59] in the early 1970s.
We may say that so-called ‘sub-Ramsey numbers’ and ‘rainbow numbers’
also fit here, but those studies went along a different track and were not
investigated as widely. We discuss the hypergraph version under the name
‘C-colouring’. Although the first published paper on this is more than forty
years old, it gained its importance only two decades ago when a more general
model was born.
In 1993 Voloshin [66], [67] introduced the concept of a mixed hypergraph
colouring, which eliminated the above asymmetry and opened up an entirely
new direction of research. Instead of H= (V, E), the basic idea is to consider
a structure H= (V, C,D), termed a mixed hypergraph, with two families of
subsets called C-edges and D-edges. By definition, a proper λ-colouring of
a mixed hypergraph H= (V, C,D) is a mapping c:V→ {1,2,...,λ}for
4
which two conditions hold:
•every C∈ C has at least two vertices of a Common colour;
•every D∈ D has at least two vertices of Different colours.
Formally, this is a combination of classical proper colourings and C-colourings,
but mixed hypergraphs were introduced independently.
The concept of a mixed hypergraph colouring has led to the discovery of
new principal properties of colourings that do not exist in classical graph and
hypergraph colourings: uncolourability in its most general setting, chromatic
polynomials of degree less than n, phantom vertices, gaps in the chromatic
spectrum, and hypergraph perfection, to name just a few. Moreover, it
led to introduction and study of new classes of hypergraphs, and further
types of constraints that are imposed on the edges of a hypergraph when
we colour the vertices. It also brings a new look at some classical graph
and hypergraph colouring problems, such as colouring planar hypergraphs,
chromatic polynomials, edge-colourings, colouring and probability, colouring
algorithms, and so on. Finally, it has generated, and continues to generate, a
wide variety of combinatorial problems and applications that formerly had no
analogy. (The newest, and perhaps most unexpected, application of mixed
hypergraphs is the modelling of problems arising in distributed computing
and cybersecurity – see [37].) A significant number of subsequent new ideas,
results and publications have led to a situation where colouring theory as a
whole is taking a new shape.
In the next section we describe a collection of significant results in classical
hypergraph colouring, where monochromatic edges are excluded. We then
consider C-colourings, which may be viewed as the counterpart by excluding
multicoloured edges. In later sections we survey some of the new trends
which, in their turn give rise to new challenges in this fast-developing area. In
particular, Section 4 is devoted to mixed hypergraphs, which opened up this
new dimension of hypergraph colouring. For further information on these, we
refer to the research monograph [68] and the regularly updated website [69].
Some results and many open problems are collected in the surveys [64] and
[2]. Finally, we discuss an even more general colouring model of hypergraphs,
introduced in 2007 by Bujt´as and Tuza [15], [13]. In that setting, lower and
upper bounds on the number of different colours, and those on the size of
the largest monochromatic vertex-subset, can be prescribed for each edge
5
independently. A hypergraph with these colouring restrictions is called a
stably bounded hypergraph.
2. Proper vertex- and edge-colourings
In this section, we survey the most significant results in classical hypergraph
colouring, their relationship with graph colouring, and some further general
hypergraph concepts and parameters.
We mention first that, for each k≥2, it is NP-complete to decide in
general whether a hypergraph is k-colourable. It remains NP-complete for
k= 2 on 3-uniform hypergraphs (see Lov´asz [47]). For all k≥2 and r≥2,
it is NP-complete also on r-uniform hypergraphs in which any two edges
share at most one vertex, except for bipartite graphs when k=r= 2 (see
Phelps and R¨odl [55]).
In fact, the decision problem for bicolourability had already been proved
to be NP-hard in 1972 by the following theorem of Woodall [73]. Let G
be a graph, and construct a hypergraph Has follows: each vertex of H
corresponds to an edge of G, and each odd cycle of Gforms an edge of H.
Then Gis 4-colourable if and only if His 2-colourable.
For uniform hypergraphs, a sufficient condition for bicolourability can
be given in terms of the number of edges. Erd˝os [29] observed that every r-
uniform hypergraph with at most 2r−1edges is bicolourable: in fact, assigning
one of two colours to each vertex randomly and independently, we obtain a
proper 2-colouring with positive probability. This bound on the number of
edges is not tight and has been improved a couple of times. The current
record is held by Radhakrishnan and Srinivasan [56].
Theorem 2.1 For sufficiently large values of r, every r-uniform hypergraph
with at most 0.7×2rpr/ ln redges is bicolourable, and can be properly 2-
coloured by a polynomial-time algorithm.
On the other hand, for every r≥2, non-bicolourable r-uniform hyper-
graphs exist with fewer than r22r+1 edges (see Erd˝os [30]).
Hypergraphs that are r-uniform and r-regular are interesting combinato-
rial objects – for instance, they describe the neighbourhood structures (both
open and closed) of regular graphs. Concerning their bicolourability, Alon
6
and Bregman [1] proved the sufficient condition r≥8, and it was recently
proved by Henning and Yeo [36] that r≥4 is sufficient. (The Fano plane
shows that the assertion is not valid for r= 3.)
Acycle of length s(s≥2) in a hypergraph H= (V, E) is a sequence
v0E0v1E1v2...vs−1Es−1vswith v0=vs, consisting of sdistinct vertices and
sdistinct edges, such that vi, vi+1 ∈Eifor all i= 0,1,...,s−1. Depending
on the parity of s, the cycle is called odd or even. Moreover, the odd cycle
v0E0v1E1v2. . . vs−1Es−1vsis an anti-Sterboul cycle if any two non-consecutive
edges are disjoint and if |Ei∩Ei+1|= 1 for every i= 0,1,...,s−2 (but not
necessarily for Es−1∩E0). A Sterboul hypergraph is a hypergraph containing
no anti-Sterboul cycle.
The next two theorems give sufficient conditions for bicolourability in
terms of conditions imposed on cycles. The first one, due to Fournier and
Las Vergnas [33], was the deepest theorem of this kind for several decades.
Its formulation with 3-critical hypergraphs is taken from Zykov [76]. (As
for graphs, a k-chromatic hypergraph is critical if the removal of any edge
decreases the chromatic number.)
Theorem 2.2 If every cycle of odd length in a hypergraph Hcontains three
edges with a vertex in common, then His bicolourable. Equivalently, every
3-chromatic critical hypergraph contains an odd cycle in which no three edges
have a vertex in common.
This theorem was recently strengthened by D´efossez [24], who proved an
old conjecture of Sterboul.
Theorem 2.3 Every Sterboul hypergraph is bicolourable.
There are also lower bounds on the number of edges in critical hyper-
graphs. The first one was proved by Seymour [58] in 1974; the second is
more recent, by Kostochka and Stiebitz [40].
Theorem 2.4 If a hypergraph on nvertices is 3-chromatic critical, then it
has at least nedges.
Theorem 2.5 If a hypergraph on nvertices is (k+ 1)-chromatic critical
and contains no 2-element edges, then the number of its edges is at least
(k−3/3
√k)n.
7
Let H= (V, E) be a hypergraph, and let Tbe a subset of V. Then the
hypergraph H/T = (T, E′) is the restriction of Hto T, where E′consists of
all non-empty intersections of edges of Ewith T. A hypergraph is said to
be balanced if every odd cycle has an edge that contains three vertices of the
cycle. Berge [7] proved the following characterization.
Theorem 2.6 A hypergraph is balanced if and only if every restriction is
2-colourable.
Let H= (V, E) be a hypergraph. The chromatic index χ′(H) of His the
least number of colours necessary to colour the edges of Hin such a way that
any two intersecting edges have distinct colours.
For a vertex v, the number of all edges containing vis the degree of v,
denoted by d(v). The maximum vertex-degree in a hypergraph H= (V, E) is
∆(H) = maxv∈Vd(v). For any hypergraph H, it is clear that χ′(H)≥∆(H).
A hypergraph Hhas the edge-colouring property if χ′(H) = ∆(H). In this
connection, the following famous result was proved by Baranyai [6].
Theorem 2.7 If nis a multiple of r, then χ′(Kr
n) = ∆(Kr
n) = n−1
r−1.
In a hypergraph H= (V, E), for any subfamily F ⊆ E we call the hyper-
graph HF= (V, F) the partial subhypergraph of H. We say that a hypergraph
His normal if every partial subhypergraph of Hsatisfies the edge-colouring
property. The following result was proved by Fournier and Las Vergnas [33].
Theorem 2.8 Every normal hypergraph is bicolourable.
A subset T⊆Vis called a transversal of a hypergraph H= (V, E) if
|T∩E| ≥ 1 for each edge E∈ E.The cardinality of a minimum transversal
is denoted by τ(H).In a hypergraph H, a set of edges that pairwise have no
vertices in common is called a matching. The maximum size of a matching
(over all matchings) is denoted by ν(H).
Since any matching is a set of pairwise non-intersecting edges, any transver-
sal must have at least one vertex from each edge of the matching. This implies
that τ(H)≥ν(H) for any hypergraph H. We say that Hsatisfies the K¨onig
property if τ(H) = ν(H). The next theorem, due to Lov´asz [46], is sometimes
called the perfect graph theorem because it implies that the complement of
a perfect graph is perfect. The latter statement was formerly known as the
weak perfect graph conjecture, formulated by Claude Berge in the early 1960s
(see Chapter 7).
8
Theorem 2.9 A hypergraph His normal if and only if every partial sub-
hypergraph H′has the K¨onig property.
Besides normal and balanced hypergraphs, there are many other classes
of hypergraphs that are either bicolourable or generalize the bipartite graphs:
hypergraphs without odd cycles, unimodular hypergraphs, hypertrees (also
called arboreal hypergraphs), Mengerian hypergraphs, paranormal hyper-
graphs, and others. The hierarchy of these and further related classes is
exhibited in [8, p.163]. Some of them have min-max properties arising from
bipartite graphs and are used in polyhedral optimization (see [8], [27]).
It may appear that increasing both the size of edges and the length of the
shortest cycle decreases the chromatic number of a hypergraph and even-
tually makes it bicolourable. This is not true: there exist high-chromatic
hypergraphs for any values of these parameters. The following theorem was
first proved by Erd˝os and Hajnal [31], and constructions were given later by
Lov´asz [45] and by Neˇsetˇril and V. R¨odl [52].
Theorem 2.10 For any integers r, s, t ≥2, there exists an r-uniform hy-
pergraph Hwith no cycles shorter than s, for which χ(H)≥t.
For a hypergraph H= (V, E), we define the bipartite (K¨onig)representa-
tion of Hto be the bipartite graph B(H) with partite sets Vand E,where a
vertex v∈Vis adjacent to a vertex E∈ E in B(H) if and only if the vertex
v∈Vis incident to the edge E∈ E in H.
A hypergraph H= (V, E) is said to be planar if B(H) is a planar graph
(see [68], [76]). This means that any planar hypergraph can be drawn in
the plane in the following way: vertices are points; each edge is a closed
curve whose interior region contains the points of the edge, and any two
edges intersect only at small neighbourhoods of their common vertices. The
remaining connected regions of the plane form the faces of such a plane
embedding. Planar graphs are special cases of planar hypergraphs: we can
replace the curves corresponding to graph edges with closed curves encircling
the adjacent vertices.
We next cite two results, the first by Bulitko [76, p.138], and the second
by Burshtein and Kostochka [76, p.138].
Theorem 2.11 The four-colour theorems for planar graphs and for planar
hypergraphs are equivalent.
9
Theorem 2.12 If a planar hypergraph contains at most one edge of size 2,
then χ(H)≤2.
There are several types of more restrictive hypergraph colourings that are
regularly encountered in the literature (see [8], [76], [68]). Here we list some
of them.
Strong colouring
Astrong λ-colouring of His a partition of Vinto λstable sets Si(i=
1,2,...,λ) such that, for each edge Ejand for every ithe inequality |Ej∩
Si| ≤ 1 holds. The strong chromatic number γ(H) is the smallest number λ
for which there exists a strong λ-colouring of H.It follows that γ(H)≥χ(H),
because every strong colouring is also a weak colouring. Note that the strong
and weak colourings coincide when His a graph. Also, γ(H) is the chromatic
number of the 2-section graph of H– that is, the graph obtained from Hon
replacing each edge Ejby the complete graph with vertex-set Ej.
Equitable colouring
An equitable λ-colouring of His a partition of Vinto λstable sets Si
(i= 1,2,...,λ) such that, for each edge Ejand for each i, the following
inequalities hold:
⌊|Ej|/λ⌋ ≤ |Ej∩Si| ≤ ⌈|Ej|/λ⌉.
Good colouring
Agood λ-colouring of His a partition of Vinto λstable sets Si(i=
1,2,...,λ) such that each edge Ejhas min{|Ej|, λ}colours. If λ≥maxj∈J|Ej|,
then a good λ-colouring is a strong λ-colouring. For each λ, any equitable
λ-colouring is also a good λ-colouring.
I-regular colouring
10
For each edge Ejin H, let ajand bjbe integers with 0 ≤aj≤bj<|Ej|.
An I-regular λ-colouring of His a partition of Vinto λstable sets Si(i=
1,2,...,λ), in such a way that, for each edge Ejand each i= 1,2,...,λ
we have aj≤ |Ej∩Si| ≤ bj.Notice that each weak colouring is an I-regular
colouring if aj= 0 and bj=|Ej|−1,that every strong colouring is an I-regular
colouring with aj= 0 and bj= 1,and that every equitable λ-colouring is an
I-regular colouring with aj=⌊|Ej|/λ⌋and bj=⌈|Ej|/λ⌉.
List-colouring
As for graphs, list-colouring and related notions (see Chapter 6) can also be
defined for hypergraphs; for example, it is also true that the choice number
of any hypergraph is equal to its fractional chromatic number [48]. Since a
hypergraph can represent a set system of any generality, this theorem implies
the corresponding equality for many kinds of colourings, including edge- and
total-colourings, generalized colourings defined in terms of hereditary prop-
erties, and so on.
There are further connections between hypergraph colourings and ex-
tremal problems, Kneser’s problem, Ramsey-type problems, etc., and many
applications. These issues are not discussed here; for information we refer to
[8] and [27].
3. C-colourings
The classical proper colouring of hypergraphs discussed in the previous sec-
tion requires that two vertices have different colours inside each edge. A
C-colouring of a hypergraph applies an opposite colouring constraint, pre-
scribing the presence of two vertices with a Common colour inside each edge.
Assigning the same colour to all vertices always yields a C-colouring. Thus,
the essential parameter here is not the minimum possible number of colours,
but the maximum. This is called the upper chromatic number of a hyper-
graph Hand is denoted by ¯χ(H).
Historically, the upper chromatic number appears in works of Sterboul
[59], Berge [8] (who called ¯χ(H) +1 the ‘cochromatic number’), Voloshin [67]
and other authors. Recent results are surveyed in [20].
The following are some basic facts concerning C-colourings.
11
•The inequality ¯χ(H)≤α(H) holds for every hypergraph: indeed, select
one vertex from each colour class of a ¯χ-colouring of H. The ¯χ-element
vertex-set obtained in this way is independent, as otherwise we would
have a polychromatic edge.
•Denote by τ2(H) the minimum cardinality of a ‘2-transversal’, a vertex-
set T⊆Vfor which |T∩E| ≥ 2 for each E∈ E. We obtain a C-
colouring of Hby assigning colour 1 to each element of Tand colour-
ing the remaining vertices with colours 2,3,...,n −τ2+ 1 pairwise
differently. This proves that ¯χ(H)≥n−τ2(H) + 1.
•Let S= (V, E) be a hyperstar, so the intersection TE∈E Eis not empty.
In addition, assume that |E| ≥ 3 for all E∈ E, and consider a vertex
zcontained in all edges. Then on deleting zfrom each E∈ E we
obtain a hypergraph S−for which ¯χ(S) = α(S−) + 1. Moreover, if S
is 3-uniform, then S−is simply a graph. It is thus easy to see that the
determination of the upper chromatic number is an NP-hard problem,
even for the class of 3-uniform hypertrees. Another consequence of this
simple construction is that the difference α(S)−¯χ(S) can be arbitrarily
large, since α(S) = |V| − 1.
In connection with the upper chromatic number, Voloshin [67] introduced
the notion of a C-perfect hypergraph. Analogously to the well-known defi-
nition of a perfect graph, we start with an inequality that is valid for all
objects and prescribe that it hereditarily holds with equality. More precisely,
let H= (V, E) be a hypergraph, and let Tbe a subset of V. Then the
hypergraph H′= (T, E′) is called the induced subhypergraph if E′consists of
all edges of Ethat lie entirely in T. A hypergraph His called C-perfect if
¯χ(H′) = α(H′) for every induced subhypergraph H′⊆ H; otherwise, His
C-imperfect. Finally, C-imperfect hypergraphs, all of whose proper induced
subhypergraphs are C-perfect, are called minimally C-imperfect.
The following are some basic facts concerning C-perfect and C-imperfect
hypergraphs.
•A typical example of a C-perfect hypergraph is a hyperstar Hwith
|TE∈E E| ≥ 2. Indeed, if His of order n, then α(H) = ¯χ(H) = n−1,
and every induced subhypergraph of it is either edgeless or has at least
two vertices contained in the common intersection of the edges.
12
•Amonostar His a hyperstar with |TE∈E E|= 1. It is always C-
imperfect, since ¯χ(H)< α(H) = n−1. (For example, the hypergraphs
in Fig. 1 are monostars.)
K1⊕K3K1⊕2K2K1⊕P4K1⊕C4
Fig. 1 The four minimal C-imperfect 3-uniform monostars
•Acycloid Cr
nis an r-uniform circular hypergraph with nvertices and
nedges (where n > r ≥3) for which each edge contains rconsecutive
vertices of the host cycle. (See Fig. 2, which is the cycloid C3
5.) Ev-
ery cycloid of the form Cr
2r−1(with r≥3) is minimally C-imperfect,
since α(Cr
2r−1) = 2r−3 and ¯χ(Cr
2r−1) = 2r−4; moreover, each of its
non-edgeless proper induced subhypergraphs is a C-perfect hyperstar.
Otherwise, Cr
nis C-perfect if n≤2r−2, and C-imperfect (but not
minimally) if n≥2r.
Fig. 2 The cycloid C3
5
Clearly, a hypergraph is C-perfect if and only if it has no induced sub-
hypergraph that is minimally C-imperfect. For some structural subclasses of
13
hypergraphs, forbidden subhypergraph characterizations have been proved
for C-perfectness. The following statements appear in [23], [18] and [19],
respectively.
Theorem 3.1
1. An interval hypergraph is C-perfect if and only if it contains no induced
monostar.
2. Let Hbe a circular hypergraph in which no edge contains any other
edge as a subset. Then His C-perfect if and only if it contains no induced
monostar and is not isomorphic to the cycloid Cr
2r−1for any r≥3.
3. A hypertree is C-perfect if and only if it contains no induced monostar.
It has been conjectured that the number of r-uniform minimally C-imper-
fect hypergraphs is finite for each r, but in fact we do not even know whether
there is a 3-uniform minimally C-imperfect hypergraph different from the six
examples shown in Figures 1, 2 and 3. Here, C6(1,2,5) ∪C6(1,3,5) is Kr´al’s
example [41], which consists of six edges obtained by rotating clockwise the
edge on the left plus two more edges on the right (they should be in the same
vertex-set).
rotate
+
Fig. 3 A minimally C-imperfect hypergraph C6(1,2,5) ∪C6(1,3,5) that is neither
a monostar nor a cycloid
On the number of r-uniform minimally C-imperfect hypergraphs, finite-
ness has been verified in the cases where also the transversal number is
bounded. More precisely, the following result was proved by Bujt´as and
Tuza [18].
Theorem 3.2 For each fixed r≥3and t≥1, there are only finitely many
r-uniform minimally C-imperfect hypergraphs with transversal number t.
14
In the same paper, Bujt´as and Tuza characterized minimally C-imperfect
hypergraphs with transversal numbers τ= 2 and τ= 3 and constructed
r-uniform examples with τ= 2 (and also with τ= 3) for which the number
of minimally C-imperfect constructions increases without bound as r→ ∞.
On the other hand, we do not know any minimally C-imperfect hypergraphs
with τ≥4.
The C-perfection of a hypergraph may lead to efficient algorithms for
solving some problems that are NP-complete in general. For example, al-
though the determination of the upper chromatic number is an NP-complete
problem, even for just the class of 3-uniform monostars, the C-perfect hyper-
trees (even without any restrictions on edge sizes) can still be ¯χ-coloured
in polynomial time (see [19]). On the other hand, the recognition problem
for C-perfect hypertrees is coNP-complete, but it becomes polynomial-time
solvable if edge-sizes are bounded from above. Further results on time com-
plexity can be found in [43] and [19], which show that the situation is more
complicated than in the case of graph perfectness.
The minimum number of r-edges, for ¯χ < r
By the pigeonhole principle, for every r-uniform hypergraph Hany vertex
colouring with r−1 colours yields a C-colouring; thus, ¯χ(H)≥r−1 always
holds. This observation leads to the extremal problem of determining the
minimum number of r-element edges on nvertices for which ¯χ=r−1. We
denote this minimum number by f(n, r), where n≥r≥2. Alternatively, we
can define f(n, r) to be the minimum number of r-element subsets of an n-
element underlying set V, selected so that, for each r-partition (V1, V2,...,Vr)
of V, there exists a selected r-element set Sthat intersects each Viin precisely
one element.
For r= 2, f(n, 2) is the minimum number of edges in a connected graph
of order n, so f(n, 2) = n−1. Taking an n-element vertex-set Vand fixing a
vertex z∈V, we obtain a general upper bound for f(n, r) from the hyperstar
H= (V, E) with edge-set
E=E∈V
r:z∈E.
Observe that ¯χ(H) = r−1 and |E| =n−1
r−1. For a lower bound, consider
an r-uniform hypergraph H= (V, E) with ¯χ(H) = r−1, and take a set
Uof r−2 vertices. Since the graph with vertex-set V\Uand edge-set
15
{E\U:U⊆E∈ E} must be connected for every choice of U, a lower bound
on f(n, r) can be calculated. These estimates appeared first in a paper of
Sterboul [60].
Theorem 3.3 For every n≥r≥3,
2
n−r+ 2 n
r≤f(n, r)≤n−1
r−1.
The lower bound in this theorem is tight for 3-uniform hypergraphs –
that is, f(n, 3) = ⌈1
3n(n−2)⌉, for each n≥3. The simplest proof appeared
in [25], and further references to a series of papers proving the same theorem
can be found in the survey [20]. For r≥4 and nlarge, however, the lower
bound is no longer tight (see [17]).
Let us mention a further equality from [60]:
f(n, n −2) = n
2−ex(n, {C3, C4}),
where the last term is the Tur´an number for graphs of order nand girth 5.
Its determination (exact or asymptotic) has been an open problem for several
decades. The ratio of known upper and lower bounds is still about √2 for
any large n. This indicates that finding an exact general formula for f(n, k)
is a hopeless task.
Recently, Bujt´as and Tuza [17] gave an asymptotic solution:
Theorem 3.4 If n > k > 2, then:
•for all fixed kand all n,
f(n, k)≤2
n−1n−1
k+n−1
k−1n−2
k−2−n−k−1
k−2;
•for all k=o(n1/3)as n→ ∞,
f(n, k) = (1 + o(1)) 2
kn−2
k−1.
16
C-type colourings of graphs
For graphs (that is, 2-uniform hypergraphs), C-colouring imposes the simple
restriction that each connected component of the graph must be monochro-
matic, so there is little to explore in this direction. But the following defec-
tive version of C-colouring introduced by Bujt´as et al. [11] raises interesting
questions even for graphs.
For each integer k≥1, a k-improper C-colouring is a colouring of ver-
tices of a graph Gsuch that, for each vertex v, at most kvertices in the
neighbourhood N(v) receive colours different from that of v; the k-improper
upper chromatic number ¯χk-imp (G) is the maximum number of colours that
can be achieved in such a colouring. Equivalently, ¯χk-imp (G) is precisely the
maximum number of components that can be obtained from Gby deleting at
most kedges at each vertex. As proved in [11], the Nordhaus–Gaddum-type
inequality
¯χk-imp (G) + ¯χk-imp (G)≤n+ 1
holds for every k > 0 and for every graph Gof order n≥8k+ 1. Moreover,
if both Gand Ghave colourings with more than one colour, then the upper
bound is 4k+ 2, independently of n.
Some further versions of graph colouring that can be described by C-type
colouring constraints are surveyed in [20].
4. Colourings of mixed hypergraphs
We recall from the end of Section 1 that a mixed hypergraph is a triple H=
(V, C,D), where both Cand Dare set-systems over the vertex-set V; their
members C∈ C and D∈ D are termed C-edges and D-edges, respectively.
A vertex λ-colouring c:V→ {1,2,...,λ}is proper if
|c(C)|<|C|,for all C∈ C,and |c(D)|>1,for all D∈ D,
where c(Y) denotes, for any Y⊆V, the set of colours assigned by the
mapping cto the vertices in Y. We say that His a bi-hypergraph if C=D,
and for any mixed hypergraph a set in C ∩ D is called a bi-edge. Note that
17
the case C=∅(termed a D-hypergraph) corresponds to a proper vertex-
colouring in the classical sense, whereas D=∅(termed a C-hypergraph) was
discussed in the previous section with respect to C-colouring.
In our historical introduction we have already indicated briefly that mixed
hypergraphs opened up a new dimension in hypergraph colouring theory. In
this section we give more details.
Example. Consider the complete r-uniform hypergraph Kr
nof order n, and
consider all r-tuples as bi-edges. Then we may use no more than r−1
colours (because C=V
r) and no colour may occur on more than r−1
vertices (because D=V
r). Thus, for n > (r−1)2, the hypergraph does
not admit any proper colouring. In particular, for r= 2 and n= 2, the
vertex pair, which would simultaneously be a C-edge and a D-edge, leads to
a contradiction of colouring requirements.
Arising from this example, we call a mixed hypergraph colourable if it
admits at least one proper colouring, and call it uncolourable otherwise. Un-
colourable mixed hypergraphs were studied in detail by Tuza and Voloshin
[63], who presented the following construction for uncolourable hypergraphs,
based on a principle substantially different from the one above.
Example. Let G= (V, E ) be a graph with chromatic number k, and define
D=E. Moreover, let C ⊆ V
kbe the k-uniform set system whose members
are the vertex-sets of Pksubpaths in G. Then H= (V, C,D) is uncolourable,
by a theorem of Gallai and Roy (see [34], [57], [62]).
A further construction from the same paper can be thought of as ‘list-
colouring without lists’. For any instance of the list-colouring problem with
a graph G= (V, E) and lists Lvon its vertices, we can extend Vwith the set
of colours to obtain a mixed hypergraph Hon the vertex-set V∪Sv∈VLv.
The D-edges of Hare the edges of G, and the C-edges are the sets Lv∪ {v},
for all v∈V. Then His colourable if and only if Gadmits a list-colouring.
Algorithmically it is an NP-complete problem to recognize colourable
mixed hypergraphs, and so there is no hope of finding easily identifiable
obstacles against colourability. Quantitatively, only the very strong require-
ment ¯χ(C)−χ(D)≥n−3 is sufficient to imply colourability; even replacing
n−3 by n−4 yields some uncolourable hypergraphs (see Tuza and Voloshin
[63]).
The chromatic inversion of a mixed hypergraph H= (V , C,D) is the
mixed hypergraph Hc= (V, Cc,Dc), with Cc=Dand Dc=C. Nearly two
18
decades ago, Voloshin [67] asked whether there is a connection between the
colourability of Hand that of Hc. A negative answer was recently given
by Hegyh´ati and Tuza [35], who proved that it is NP-complete to test
whether the chromatic inversion of a colourable 3-uniform mixed hypergraph
is colourable, and coNP-complete to test whether the chromatic inversion
of an uncolourable 3-uniform mixed hypergraph is uncolourable. This neg-
ative result is derived from the positive theorem that if D=V
3and His
3-uniform, then the colourability of His decidable in polynomial time.
Properties of colourable mixed hypergraphs
For the rest of this section we restrict our attention to mixed hypergraphs
H= (V, C,D) that are colourable. Under this assumption it is interesting to
study the feasible set Φ(H) of H, defined as the set of those integers kfor
which Hadmits a proper k-colouring with exactly kcolours.
Among the elements of Φ(H), the minimum χ(H) and the maximum
¯χ(H) are termed the lower chromatic number and upper chromatic num-
ber, respectively. In some classes of hypergraphs, where ¯χis near to n, the
decrement, defined as dec(H) = n−¯χ(H), may be informative.
Further information, more detailed than Φ(H), about the colourability
properties of a mixed hypergraph, is provided by the chromatic spectrum.
It is defined as the sequence (r1, r2,...,rn), where each rkis the number
of feasible partitions – that is, the partitions of the vertex-set into colour
classes induced by proper colourings using precisely kcolours. A gap in
the chromatic spectrum and in the feasible set is an integer kfor which
χ(H)< k < ¯χ(H) and rk= 0.
If His a D-hypergraph, then its feasible set is an interval of integers:
Φ(H) = {χ(H), χ(H) + 1,...,n}. Similarly, if His a C-hypergraph, then
its feasible set is the interval Φ(H) = {1,2,...,¯χ(H)}. Somewhat unex-
pectedly, the feasible sets of non-1-colourable mixed hypergraphs are rather
unrestricted – namely, for any finite set S⊂N\ {1}of integers, there exists
a mixed hypergraph Hfor which Φ(H) = S(see Jiang et al. [38]). Even
more generally, for any finite sequence (rk)b
k=aof non-negative integers with
2≤a≤band ra6= 0 6=rb, there exists a mixed hypergraph Hfor which
χ(H) = a, ¯χ(H) = b, and for each a≤k≤b, the number of feasible parti-
tions is exactly rk(see Kr´al’ [42]). A similar statement is valid for feasible
sets of r-uniform mixed hypergraphs with only one additional restriction: if
an integer k < r −1 belongs to Φ, then all integers between kand r−1 also
19
have to be contained in Φ (see Bujt´as and Tuza [14]). The case of chromatic
spectra with r1= 0 and rk= 0 or 1 for all k≥2, was recently studied by
Zhao, Diao and Wang [74], [75], especially concerning the minimum number
of vertices and edges.
Hypertrees are more restrictive: colourability implies that χ= 2 (unless
D=∅, in which case χ= 1) and that the feasible set is gap-free. More
generally, no gap occurs if the hypergraph admits a host graph in which any
two cycles are vertex-disjoint (see Kr´al et al. [43]). A proper 2-colouring of a
hypertree is also easy to find: as long as there exist C-edges of size 2, contract
them to single vertices, and then colour the contracted host tree properly in
the usual sense. A hypertree is uncolourable if and only if this procedure
yields a contracted D-edge of size 1 (see Tuza and Voloshin [63]).
The upper chromatic number turns out to be a harder issue. Although
Bulgaru and Voloshin [23] proved that it can be expressed with a nice formula
on mixed interval hypergraphs, the determination of ¯χon hypertrees is NP-
hard (see [43]). Moreover, various results on its non-approximability are
known (see [22]).
For the feasible set of a planar mixed hypergraph H, Kobler and K¨undgen
[39] proved that either Φ(H) is a gap-free interval with minimum value 4 or
less, or the lower chromatic number is χ= 2 and the only gap occurs at 3.
On the boundary of colourability and uncolourability, those mixed hy-
pergraphs have been located that admit just one feasible partition – that is,
those with just one rk= 1 in their chromatic spectrum, and zeros otherwise.
The systematic study of these uniquely colourable mixed hypergraphs was
initiated by Tuza, Voloshin and Zhou [65]. This class is NP-hard to rec-
ognize (it is coNP-complete when the input hypergraph is given, together
with a proper colouring) and it does not admit a characterization in terms
of forbidden induced subhypergraphs; in fact, every mixed hypergraph can
be embedded as an induced subhypergraph into a uniquely colourable one.
Moreover, Bujt´as and Tuza [12] proved that it is also NP-complete to decide
whether Hadmits a vertex ordering v1, v2,...,vnsuch that, for all 1 ≤i≤n,
the subhypergraph induced by {v1, v2...,vi}is uniquely colourable.
Some uniquely colourable subclasses admit a good characterization and
efficient recognition algorithm – for instance, those with χ=n−1 and χ=
n−2 (see [54]), uniquely colourable mixed hypertrees (see [53]) and circular
mixed hypergraphs (see [70]). Moreover, the possible size distributions of
colour classes in uniquely colourable r-uniform bi-hypergraphs have also been
characterized (see [5]).
20
An analogy with complete graphs is that uniquely colourable separators
(vertex subsets that induce a uniquely colourable subhypergraph and whose
removal makes Hdisconnected) can be applied to derive a recursive for-
mula to compute the chromatic polynomial (see [65]). On the other hand,
colourable graphs do not satisfy the requirement put on uniquely colourable
mixed hypergraphs; instead, they form a subclass of weakly uniquely colourable
mixed hypergraphs; this means that rχ=r¯χ= 1. They have also been stud-
ied to some extent in [65].
C-perfect mixed hypergraphs
C-perfectness can be defined for mixed hypergraphs H= (V, C,D) in the same
way as for C-colouring, by requiring that ¯χ(H′) = αC(H′) for all induced
subhypergraphs H′= (V′,C′,D′) of H, where αC(H′) is the independence
number of the hypergraph (V, C′). Generalizing the class of monostars, one
can obtain C-imperfect mixed hypergraphs by taking some C-edges with non-
empty intersection, and inserting each 2-element subset of their common
intersection as a D-edge. (These constructs are called polystars.) But the
characterization problem for C-perfect mixed hypergraphs looks even harder
than that for C-perfect hypergraphs. Some examples given in [19] indicate
that the situation already becomes quite complicated for mixed hypertrees.
Steiner systems and finite projective planes
ASteiner system S(t, k, v) is a k-uniform hypergraph of order v, for which
each t-tuple of vertices is contained in precisely one edge. To mention some
celebrated examples, a system S(2,3, v) is a Steiner triple system STS(v), an
S(3,4, v) is a Steiner quadruple system SQS(v), and an S(2, q + 1, q2+q+ 1)
is a finite projective plane of order q.
The edges are traditionally called blocks. We may view each block as a
C-edge (when an STS(v) is denoted by CSTS(v)) or as a bi-edge – that is,
aC-edge and a D-edge at the same time (when an STS(v) is denoted by
BSTS(v)). The notations CS(t, k, v), BS(t, k, v), CSQS(v) and BSQS(v) are
derived for the respective systems in a similar way.
The study of the upper chromatic number in Steiner triple systems started
with a paper by Milazzo and Tuza [49], where the authors proved that
¯χ(BS T S(v)) ≤¯χ(C ST S(v)) ≤s,
21
for all v≤2s−1.This upper bound on ¯χis tight for all s≥2, and the
systems attaining equality were also characterized. More generally,
¯χ(BS(t, t + 1, v)) ≤¯χ(CS(t, t + 1, v)) = O(ln v)
holds for any fixed t≥2 as v→ ∞ (see [50]). Interestingly, from below it is
not even known whether ¯χ(BSQS(v)) can tend to infinity with v. Moreover,
no uncolourable BSQS has so far been found.
Concerning the decrements of projective planes Π(q) of order q, Bacs´o and
Tuza [4] proved that dec(Π(q)) ≥2q+√q/2−o(
√q). For an infinite sequence
of values qthis is provably optimal, even in the order Θ(√q) of its second
term: if qis a square, then the union of two disjoint Baer subplanes in the
Galois plane P G(2, q) meets each line in more than one point, and this allows
us to construct a colouring; from which we deduce that dec(P G(2, q)) ≤
2q+2√q+1. Recently, Bacs´o, H´eger and Sz˝onyi [3] gave sufficient conditions
to ensure that dec(P G(2, q)) = τ2((P G(2, q))) −1. (The right-hand side is
always an upper bound if there exists a 2-transversal of minimum cardinality
that is independent – see the section on C-colouring.)
For projective planes Π(q) without an underlying algebraic structure, a
general upper bound for dec(Π(q)) is 3q−2. Indeed, one can pick three lines
L, L′, L′′ in general position (that is, with empty intersection), and assign
colour 1 to L′∩L′′ and to all points of L\(L′∪L′′ ), and colour 2 to all
points of (L′∪L′′)\(L′∩L′′ ). Then every line contains two points of the
same colour, and so each remaining point can have its distinct colour.
More details about colourings of block designs can be found in the survey
[51].
5. Colour-bounded and stably bounded hypergraphs
In this section we consider generalizations of mixed hypergraphs; we shall
view them as hypergraphs H= (V, E) with additional constraints prescribed
for the edges. For convenience, we assume that E={E1, E2...,Em}.
These structures include colour-bounded hypergraphs, stably bounded hy-
pergraphs, and pattern hypergraphs.
The most general class is the third one, introduced by Dvoˇr´ak et al. [28].
For each edge Ei, a family Piof partitions is specified, and a vertex-colouring
is considered to be proper if its colour classes induce a partition of each Ei
which is a member of Pi. Even in this very general model, the authors
managed to develop a theory concerning feasible sets with or without gaps.
22
The other two classes are closer to the flavour of mixed hypergraphs;
their conditions on the edges are given quantitatively. For the more general
one, a stably bounded hypergraph required four functions s, t, a, b :E → N
to be given: we write si=s(Ei), ti=t(Ei), ai=a(Ei), bi=b(Ei), for all
1≤i≤m. Given these parameters, a colouring is proper if, for each Ei∈ E,
•the number of different colours assigned to the vertices of Eiis at
least si;
•the number of different colours assigned to the vertices of Eiis at
most ti;
•there exists a colour assigned to at least aivertices of Ei;
•no colour occurs on more than bivertices of Ei.
So the functions sand timpose bounds on the largest polychromatic subsets
of the edges, while the functions aand bimpose bounds on their largest
monochromatic subsets.
These four colour-bounding functions can capture many kinds of parti-
tioning problems, allowing a concise model description. A practical example
of this is resource allocation where, loosely speaking, the lower bounds (en-
suring several kinds of resources or multiple copies of one resource) increase
the security and stability of a system, while the upper bounds correspond to
constructing the system in an economical way (since more pieces of resource
increase the cost).
Functional subclasses
Colour-bounding functions of types si= 1, ti=|Ei|,ai= 1, and bi=|Ei|are
called non-restrictive on the edge Ei, because they express no real restric-
tions on proper colourings. A colour-bounding function is non-restrictive in
a hypergraph Hif it is so on all edges of H; otherwise, it is called restric-
tive. In notation, we write capital letters to indicate the functions that are
allowed to be restrictive in a hypergraph. For instance, an (S, T )-hypergraph
means that only sand tare given – that is, only the numbers of different
colours occurring in the edges are bounded; an (S, T )-hypergraph is also
called a colour-bounded hypergraph. Similarly, in S-hypergraphs only the
minimum number of required colours is given for each edge. Note that ev-
ery S-hypergraph is an (S, T )-hypergraph at the same time, because tis not
required to be restrictive for the latter.
23
Chromatic polynomials
As in the classical theory of graph colourings, for a hypergraph H= (V, E)
and an integer λ∈N, we denote by P(H, λ) the number of proper colourings
c:V→ {1,2,...,λ}. This P(H, λ) is a polynomial in λfor every H(in
the most general class, pattern hypergraphs, too), and is thus termed the
chromatic polynomial of H. Voloshin [66] extended the concept of P(H, λ)
from the classical one to mixed hypergraphs, and discovered some new prop-
erties of P(H, λ). For example, he showed that the degree of a chromatic
polynomial is not determined by the number of vertices in general, as it
is equal to the upper chromatic number ¯χ(H); only in classical graph and
hypergraph colourings do the two invariants coincide. Another unusual prop-
erty is that some mixed hypergraphs contain vertices whose removal from H
changes nothing with respect to colourability: all colourings and P(H, λ) re-
main the same. Such vertices are called phantom vertices; they are invisible
for P(H, λ), and Hmay contain any number of them.
The famous connection–contraction algorithm for finding the chromatic
polynomial, originated by Birkhoff [10] in 1912, formalized by Whitney [72] in
1932, and generalized to hypergraphs by Zykov [76] in 1974, was extended to
the splitting–contraction algorithm for mixed hypergraphs by Voloshin [66]
in 1993. Further, some properties of P(H, λ) in S-hypergraphs have been
described in [26].
It is not known which polynomials occur as chromatic polynomials of
mixed hypergraphs. For non-1-colourable structures, however, Bujt´as and
Tuza [15] gave the following necessary and sufficient conditions. Here S(n, k)
is the Stirling number of the second kind, the number of partitions of n
elements into precisely knon-empty sets.
Theorem 5.1 Let P(λ) = Pℓ
k=0 akλkbe a non-zero polynomial for which
P(1) = 0 (so Pℓ
k=0 ak= 0). Then the following properties are equivalent.
•P(λ)is the chromatic polynomial of a colour-bounded hypergraph;
•P(λ)is the chromatic polynomial of a mixed hypergraph;
•P(λ)satisfies the following conditions:
–all coefficients akof P(λ)are integers;
–the leading coefficient aℓis positive;
24
–the constant term a0is 0;
–for each positive integer j≤ℓ,Pℓ
k=jak·S(k, j)≥0.
These characterizations remain the same if we replace the pair (S, T )
(colour-bounded hypergraphs) by any of (S, A),(T, B),and (A, B) (those
that admit uncolourable hypergraphs), or by (S, T , A, B) itself, or by pattern
hypergraphs.
One way to describe relations between functional classes (defined in terms
of the subsets of {S, T , A, B}) is to consider the classes of chromatic poly-
nomials that occur in them. In Fig. 4 we exhibit their Hasse diagram; we
indicate the positions of the classes of C-, D-, and mixed hypergraphs by
PC,PDand PM, respectively. The sets of chromatic polynomials belonging
to non-1-colourable (S, T , A, B)-hypergraphs and to non-1-colourable (S, T )-
hypergraphs are the same; the difference shown in Fig. 4 arises from the fact
that there are 1-colourable (S, T, A, B)-hypergraphs whose chromatic spectra
do not occur for any (S, T )-hypergraphs. Note that there are examples of
problems (such as algorithmic problems), and of classes, where the problem
is provably harder on the class located lower in the hierarchy of chromatic
polynomials.
Ps,t,a,b =Ps,a =Pa,b
Pa=Pa,t Ps,t,b =Ps,t =Pt,b =PM
PC=PtPs=Ps,b
PD=Pb
Fig. 4 The Hasse diagram for classes of chromatic polynomials
Interval hypergraphs and hypertrees
The colouring properties of some well-structured hypergraph classes have
been analysed in detail. Many results are known about the computational
25
complexity of the problems of deciding colourability and of determining χ
and ¯χfor hypertrees, interval hypergraphs and some of their subclasses. For
a summary of what is known and what is still open, we refer to the three
handy tables in [21]. Below we mention a few of these results.
The cases of just one restrictive function are quite well understood, except
in interval A-hypergraphs, where the algorithmic complexity of determining
¯χis still an open problem. The interval S- and B-hypergraphs admit op-
timal periodic colourings, with χ= max siand χ= max⌈|Ei|/bi⌉colours,
respectively. We observe that ¯χfor interval T-hypergraphs can be computed
efficiently.
Concerning two restrictive functions, the simplest case is (S, B), for which
χ= max(si,⌈|Ei|/bi⌉). For each of the pairs (T, B),(A, B) and (S, A), how-
ever, it is NP-complete to decide whether a given interval hypergraph is
colourable. (This implies intractability for any three restrictive functions,
while colourability and the values of χand ¯χcan be determined efficiently,
even for (S, T, A, B) on classes of hypergraphs with bounded edge size.) The
important unsolved case is to decide colourability and to determine ¯χfor
an interval (S, T )-hypergraph. But interestingly enough, as proved in [16],
once we know that it is colourable, we immediately obtain χ= max si, and
if a proper colouring is given also, then it can be transformed efficiently to
an optimal one; moreover, the feasible set is gap-free. For other pairs of
restrictive functions, only a few upper bounds for χhave been found for in-
terval hypergraphs (for example, χ≤max |Ei|, even for (S, T , A, B)), and it
is not known whether some interval (S, A)-hypergraphs, (T, B)-hypergraphs
or (A, B)-hypergraphs can have any gaps between χ+ 1 and max |Ei| − 1.
Colour-bounded and stably bounded hypertrees turn out to be much more
complex than mixed hypertrees. The decision problem for colourability is
NP-complete for any non-trivial pair of restrictive functions (that is, other
than (S, B) and (T , A)), even in the 3-uniform case. Recalling that every
mixed hypertree has a gap-free feasible set, we report that a further dramatic
change is illustrated by the following theorem from [16].
Theorem 5.2 Let Fbe a finite set of positive integers. Then there exists an
(S, T )-hypertree with feasible set Fif and only if min(F) = 1 or min(F) = 2
and Fcontains all integers between min(F)and max(F), or min(F)≥3.
Finally, we mention that the results on interval (S, T )-hypergraphs can
be partially extended to circular (S, T )-hypergraphs (see [16]); namely, if
26
a hypergraph from this class is colourable, then it satisfies the inequality
χ≤2(max si)−1 and no gaps can occur between 2(max si)−1 and ¯χ.
For max si≥3 we do not know, however, whether there can exist any gaps
between χand 2(max si)−1.
6. Conclusion
In this chapter we described the current state of hypergraph colourings as a
separate subject. With so many relations to graph colouring and new fun-
damental ideas, results and applications, this area is developing fast. The
impact of every result may be to give rise to a set of new problems, often with-
out any analogues from the past. While the number of publications continues
to grow, we observe that some possible subdirections remain untouched. We
predict that hypergraph generalizations of such areas as colouring graphs
on surfaces, edge-colourings, colouring and probability, Hadwiger’s conjec-
ture, colouring games, orientations and flows (to name just a few) have great
research potential in the observable future.
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