Review articleIn praise of homomorphisms☆
Introduction
Given two (undirected) graphs ) a homomorphism of to is a mapping satisfying for every .
Thus homomorphisms are just mapping which preserve all edges (and in general nothing else). This notion present a natural habitat to study graphs particularly from the algebraic point of view. Homomorphisms generalize isomorphisms, subgraphs, embeddings, but also colorings and various other special partitions.
Being in nature algebraic, homomorphisms generalize to other structures such as relational structures or even more finite models which are specified by a language . This can be done as follows: Let be a set containing relational symbols (like ) and function symbols (like ). Each relational symbol comes with arity which is a natural number. Each function symbol comes with domain arity (in this paper the range arity can be assumed without loss of generality equal to ). An - is then a set A together with relations for every relation symbol and together with functions for every function symbol . Such -structures are also called . We are interested in the case that all sets are finite. If contains no function symbols then we speak about structures.
Given two -structures and a homomorphism to is any mapping satisfying: and
Such notion is very flexible and it is domain of universal algebra, theory of categories and model theory (to name just a few). This was also the original context in which homomorphisms were studied (see e.g. [1], [2]). But
at several occasions the attention turned to combinatorial side thus displaying the emerging maturity of graph theory and (early) theoretical computer science.
One should mention here pioneering works of Sabidussi, Hedrlín and Pultr. These papers had not only a direct influence but perhaps more importantly (and more indirectly) encouraged more algebraical (or categorical) reasoning. This history is outlined in the first book devoted to this area Hell, Nešetřil [3] (and partly also in Godsil, Roy [4]). Here we complement this by providing a personal outline of some more recent results and problems which seem to be motivating today research.
The development could be described along the following key words:
Invariants, Constructions, Model Theory, Complexity, Data Science.
We have to be selective and thus we cover here just three of those areas and we have to be brief even then. Note that presently authors prepare the second edition of [3] and that Lovász [5] contains a very nice chapter on homomorphisms. The homomorphisms are also treated in [6] from different perspective of sparsity. Among other things which are not covered here are Ramsey theory [7], [8] and e.g. Rossman’s Homomorphism preservation theorem [9].
Section snippets
Invariants
Homomorphism related invariants are abundant. Particularly the homomorphism numbers have a great relevance here:
for graphs (or more generally -structures) we denote by the set of all homomorphisms of to . By we denote the number of all homomorphisms of to . Thus .
The function is an invariant which particularly describes isomorphisms:
This is a classical (and famous) result of Lovász
Constructions
Instead of real valued invariants we often consider 0–1 invariants and decision problems. For example instead of considering all homomorphisms from to we just ask whether there exists such a homomorphism (which in the case is the question whether a given graph is -colorable). In the other words, instead of considering the category of graphs and all homomorphisms between them we consider a simplification of this category which is a quasiorder. The categorical context then provided
Complexity
Complexity is of course holy grail of theoretical computer science. These days most combinatorial a graph theory results have to be studied as well from complexity point of view. And in this context the study of chromatic number stands out as a particularly intensively studied “hard” problem. Coloring of graphs with special properties and special coloring of graphs is thus both classical and ever inspiring part of our research. By writing this survey we want in this context just to mention
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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