Considering color-bounded hypergraphs with restricted structures, first we deal with interval hypergraphs and point out some special coloring properties. Throughout this section we assume a hostpathgraph with vertex orderx1, x2, . . . , xn. Since there exists a unique path from xi to xj for every 1 ≤ i ≤ j ≤ n, we shall write [xi, xj] for the ‘closed interval’ and, analogously, the notation for the ‘open’ and ‘half-open’
intervals will be used, too, as introduced at the beginning of this chapter.
It was proved in [29] that no gaps can occur in the chromatic spectrum of any mixed interval hypergraph. Our first theorem generalizes this result.
Theorem 14. Every colorable interval hypergraph I = (X,E,s,t) has a gap-free chromatic spectrum, and its lower chromatic number is equal to s= maxEi∈Esi. Proof Trivially, if I is colorable, then its lower chromatic number is at least s.
Therefore, to prove both statements of the theorem, it is enough to show that any
k-coloring of I can be transformed to a proper (k −1)-coloring, whenever k > s holds. (This recoloring process is illustrated by an example on Figure 6.)
Consider a k-coloring ϕ of I (k > s) and determine the following two vertices.
Let a be the vertex for which [x1, a] is the inclusion-wise minimal starting interval containing all thek colors, that is |ϕ[x1, a]| =k and ϕ(a)∈/ ϕ[x1, a[. Then counting the different colors backwards from a in the ordering, choose the vertex b for which
|ϕ[b, a]|=s+ 1 and|ϕ]b, a[|=s−1.
Next, we show that the conditions of the Recoloring Lemma are fulfilled by A= [x1, b], B =]b, a[, C = [a, xn], α =ϕ(a) and β =ϕ(b).
• Since α /∈ϕ[x1, a[ and β /∈ϕ]b, a[, the conditions (1) and 2(b) are satisfied.
• If a hyperedge Ei meets both A and C, then it surely contains the interval [b, a]. Hence, for this edge |ϕ(Ei ∩ B)| = s −1 and α ∈ ϕ(Ei ∩C) hold, complying with 2(c) and 2(a).
Consequently, the Recoloring Lemma can be applied, assuring that the transpo-sition of colors αand β on the intervalC yields a proper coloring ϕ′ ofI. After this recoloring the interval [x1, a] has only k−1 colors, and either a (k−1)-coloring is obtained, or we still have a k-coloring for which a similar recoloring can be applied again. In the latter case, however, the cardinality of the set C is smaller than it was in the previous recoloring, hence the repeated application of this procedure yields a (k−1)-coloring ofI in a finite number of steps.
Starting with k= ¯χ(I), the number of colors can be decreased one by one, thus a coloring is generated for all k ∈ {s, s+ 1, . . . ,χ(¯ I)}. This completes the proof of
the theorem.
Remark 12.
1. Having a k-coloring (k > s) as input of the above algorithm, the number of vertices in [x1, a[ increases by at least one in each recoloring, until the selected color gets eliminated. Therefore it needs at most O(n) phases of recoloring to generate a (k −1)-coloring. Moreover, based on the previous proof, one phase can be implemented in O(n) time. Hence, it takes O(n2) steps to get a (k−1)-coloring.
2. If our goal is to obtain a χ-coloring from a given k-coloring (k > χ=s), the above algorithm can be slightly modified. In this case in each recoloring phase let a be the vertex for which [x1, a] is the inclusion-wise minimal starting interval containing s+ 1 different colors. In this way, we can directly obtain a χ-coloring in O(n2) time.
Evidently, every interval [s, t] of positive integers is a feasible set of some color-bounded interval hypergraph. (E.g., consider a hypergraph with only one (s, t)-edge containing all the at leasttvertices.) Combining this with Theorem 14, the following characterization is obtained:
Corollary 13. For a set S of positive integers there exists a color-bounded interval hypergraph I whose feasible set is Φ(I) = S if and only if S is an interval of integers.
There is an efficient algorithm formixed interval hypergraphs to test their col-orability, and also to compute their upper chromatic number [16]. But when the colorings ofcolor-bounded interval hypergraphs are considered in general, this seems to be a more difficult problem. Nevertheless, there are some particular types of in-terval hypergraphs for which we can design polynomial-time algorithms to decide whether they are colorable.
Proposition 21. If I is an interval hypergraph satisfying the further condition s = max
Ei∈E si ≤ min
Ei∈E ti =t then I is colorable and χ(¯ I)≥t.
Proof For any s≤ k ≤t, let the vertex xi get color i (modk) for i= 1,2, . . . , n.
This periodical coloring is proper for I, because each Ei gets precisely k colors, due
to the assumptions si ≤s≤k ≤t≤ti ≤ |Ei|.
Let us note that mixed interval hypergraphs do belong to this class after the contraction of C-edges of size 2, provided that no D-edge of size 1 arises (that is, the original hypergraph does not contain an obvious obstruction against colorability).
Proposition 22. The colorability of interval hypergraphs without edges of more than three vertices can be decided in linear time.
Proof First, we contract each (1,1)-edge to one vertex, and then we check whether every edge withsi =j (j = 2,3) contains at least j vertices. If this trivial necessary condition holds, then the contracted (and hence also the original) hypergraph is colorable. For example, one can get a proper coloring by the following procedure:
Letϕ(x1) = 1 andϕ(x2) = 2; and for i= 3, . . . , n let ϕ(xi) =ϕ(xi−2) unless xi
is the last vertex of a (3,3)-edge. In the latter case, letxi get a third color, which is different from both previous ones ϕ(xi−1) and ϕ(xi−2). Then every non-(3,3)-edge of the contracted hypergraph gets precisely two colors, hence the coloring is proper.
Here we only mention, without proof, that there is a greedy coloring algorithm also for another type of color-bounded interval hypergraphs. The constraint is that
any two hyperedges ofI should be disjoint or one of them should contain the other.
It can be decided in polynomial time whether such a hypergraph is colorable, and if it is, the algorithm computes the upper chromatic number χ(I), and produces a χ(I)-coloring.