The new structure classes studied in the Thesis provide a general framework for modeling problems from real life. In Chapter 9, concentrating on the field of infor-matics, we discuss several possible applications in detail.
By definition, in a stably bounded hypergraph we can prescribe the number of colors occurring inside each hyperedge and we can take bounds for the cardinality of the largest monochromatic subset of each edge. But in many applications we have restrictions concerning the number of occurrences of fixed types; that is, of fixed colors. It is shown that a stably bounded hypergraph can be supplemented with new vertices (corresponding to the colors) and edges such that the obtained hypergraph expresses the above type of constraints, too.
• The resource allocation problem appears in informatics in several forms. This means a mapping of tasks or processing. The requirements on the (in)compatibility and the number of occurrences can be efficiently expressed using stably bounded hypergraphs.
• Problems in connection with dependability and fault tolerance of IT systems can also be modeled by stably bounded hypergraphs. Here the typical con-straints concern the least number of identical resources (replicas) to assure a given level of fault tolerance, whilst an upper bound expresses a cost limit.
• In general sense, the frequency assignment problem means assigning frequen-cies to the transmitters so that excessive interference is avoided. This problem appears in different forms when concerns mobile telephone networks, radio and television broadcasting, or satellite communication. These different forms yield different constraints for frequency assignment and hence, they inspired different non-classical versions of graph coloring such as distance-labeling, T-coloring, and their more general versions. We will point out that all these varieties can be described in a unified and natural way, using stably bounded hypergraphs.
• Furthermore, we give examples of possible applications regarding the fields of service-oriented architecture, scheduling of file transfers, and data access in parallel memory.
It is worth noting that in the monograph [59] some applications of mixed hyper-graphs are discussed from the fields of molecular biology and genetics of populations.
Concerning the application of S-hypergraphs there can be found economical exam-ples in the paper [19].
2 Feasible sets of uniform mixed hypergraphs
Concerning graph and hypergraph coloring in the classical sense, we have only one type of constraints. Namely, no edge can have its endpoints with a common color and no hyperedge can have all its vertices with the same color. These requirements can easily be satisfied, since we trivially obtain a proper coloring if every vertex is labeled with a dedicated color. On the other hand, if we have a proper coloring with fewer than n = |X| colors, then any color class containing more than one vertex can be split into two non-empty parts and the coloring conditions remain fulfilled.
Thus, for any k between the (lower) chromatic number and n there exists some coloring using precisely k colors. This results in a transparent structure of possible feasible sets. For any positive integers 2≤a≤b, we can easily construct graphs or hypergraphs which havek-colorings if and only ifa≤k ≤b. This situation does not change essentially even if it is prescribed that every hyperedge is of the same size r.
(More precisely, the only additional condition isa ≤ ⌊r−1n ⌋=⌊r−1b ⌋.) But the above simple structure of traditional colorings can become disadvantageous when we have to model a problem with a more complex system of constraints.
The notion of mixed hypergraph allows the usage of two opposite conditions:
we can require for some fixed groups (C-edges) that each of them should contain two elements labeled identically, while the traditional coloring constraint concerns the D-edges; that is, the latter have to contain two elements labeled differently.
By the simultaneous presence of C- and D-edges, a more complex structure with surprising features is obtained. A mixed hypergraph can be uncolorable; or, if it is colorable, the possible numbers of used colors may not form a ‘continuous’ interval at all (i.e., in their feasible sets there can occur an unrestricted number of ‘gaps’
with unrestricted sizes). On the one hand, these properties indicate the fact that mixed hypergraphs are applicable for modeling a wide range of practical problems.
On the other hand, it is important to study in subclasses of mixed hypergraphs, whether these ‘irregular’ properties can occur on their members. These results can help in the selection of an appropriate model for a given practical problem. Here we will characterize the possible feasible sets of r-uniform mixed and bi-hypergraphs, answering two open questions of this field.
2.1 Characterization theorem
It is readily seen that if 1 ∈ Φ(H), then H cannot have any D-edges, therefore in this case the feasible set Φ(H) necessarily is gap-free; and vice versa, any gap-free
‘interval’ {1, . . . , k} is a feasible set of some mixed (C-) hypergraph. On the other hand, for the case 1 ∈/ Φ(H), Jiang et al. proved in [29] that for any finite set S of integers greater than 1 there exists a mixed hypergraph H with Φ(H) = S.
But the corresponding problem for bi-hypergraphs in general and r-uniform mixed hypergraphs was open for several years.
In this chapter we consider r-uniform mixed hypergraphs, i.e. those with |C|=
|D| = r for all C ∈ C and all D ∈ D, with a fixed integer r ≥ 3. Our main result regarding possible feasible sets is the following characterization:
Theorem 1. Let r ≥3be an integer, and S a finite set of natural numbers. There exists a colorable r-uniform mixed hypergraph H with Φ(H) =S and |C|+|D| ≥1 if and only if
(i) min(S)≥r, or
(ii) 2≤min(S)≤r−1 and S contains all integers between min(S) and r−1, or (iii) min(S) = 1 and S ={1, . . . ,χ¯} for some natural number χ¯≥r−1.
Moreover, S is the feasible set of some r-uniform bi-hypergraph with C =D 6=∅ if and only if it is of type (i) or (ii).
If r = 3, then (i) and (ii) together allow any set not containing the element 1.
Hence, a characterization for bi-hypergraphs can be concluded.
Corollary 1. A finite set S of positive integers is the feasible set of some bi-hypergraph with at least one bi-edge if and only if 1∈/S.
Moreover, we obtain
Corollary 2. For every mixed hypergraph H with χ(¯ H) > 1 there exists a 3-uniform mixed hypergraph H3 such that Φ(H) = Φ(H3); and if 1∈/Φ(H), then H3 can be chosen as a bi-hypergraph.
Remark 1. Deleting the condition |C|+|D| ≥ 1 from Theorem 1 (i.e. admitting mixed hypergraphs and bi-hypergraphs without hyperedges), we obtain that S is a feasible set of an r-uniform mixed hypergraph if and only if S satisfies
(i) or (ii) or
(iii)′ min(S) = 1 and S ={1, . . . ,χ¯} for some natural number χ.¯ The same is true for feasible sets of r-uniform bi-hypergraphs.
We begin with some easy observations, on fewer thanrcolors, in Section 2.2. The essential part of the proof of Theorem 1 is split into three sections. In Section 2.3 we construct bi-hypergraphs whose feasible sets contain just one element larger than r−1. Then in Section 2.4 we show how to combine them in order to generate a feasible set with an unrestricted number of prescribed elements larger than r−1, but still admitting an (r−1)-coloring. The feasible sets having no elements smaller than r are treated in Section 2.5.