Jaroslav Nešetřil

On the complexity of H -coloring

Abstract Let H be a fixed graph, whose vertices are referred to as ‘colors’. An H-coloring of a graph G is an assignment of ‘colors’ to the vertices of G such that adjacent vertices of G obtain adjacent ‘colors’. (An H-coloring of G is just a homomorphism G → H). The following H-coloring problem has been the object of recent interest: Instance: A graph G. Question: Is it possible to H-color the graph G? H-colorings generalize traditional graph colorings, and are of interest in the study of grammar interpretations. Several authors have studied the complexity of the H-coloring problem for various (families of) fixed graphs H. Since there is an easy H-colorability test when H is bipartite, and since all other examples of the H-colorability problem that were treated (complete graphs, odd cycles, complements of odd cycles, Kneser graphs, etc.) turned out to be NP-complete, the natural conjecture, formulated in several sources (including David Johnsons NP-completeness column), asserts that the H-coloring problem is NP-complete for any non-bipartite graph H. We give a proof of this conjecture.

Otakar Boruvka on minimum spanning tree problem translation of both the 1926 papers, comments, history

Abstract Borůvka presented in 1926 the first solution of the Minimum Spanning Tree Problem (MST) which is generally regarded as a cornerstone of Combinatorial Optimization. In this paper we present the first English translation of both of his pioneering works. This is followed by the survey of development related to the MST problem and by remarks and historical perspective. Out of many available algorithms to solve MST the Borůvkas algorithm is the basis of the fastest known algorithms.

The core of a graph

Abstract The core of a graph is its smallest subgraph which also is a homomorphic image. It turns out the core of a finite graph is unique (up to isomorphism) and is also its smallest retract. We investigate some homomorphism properties of cores and conclude that it is NP-complete to decide whether or not a graph is its own core. (A similar conclusion is reached about testing whether or not a graph is rigid, i.e., admits a non-identity homomorphism to itself.) We also give a polynomial-time verifiable condition for a graph of small independence number to be its own core.

Tree-depth, subgraph coloring and homomorphism bounds

We define the notions tree-depth and upper chromatic number of a graph and show their relevance to local-global problems for graph partitions. In particular we show that the upper chromatic number coincides with the maximal function which can be locally demanded in a bounded coloring of any proper minor closed class of graphs. The rich interplay of these notions is applied to a solution of bounds of proper minor closed classes satisfying local conditions. In particular, we prove the following result:For every graph M and a finite set F of connected graphs there exists a (universal) graph U = U(M, F) ∈ Forbh(F) such that any graph G ∈ Forbh(F) which does not have M as a minor satisfies G → U (i.e. is homomorphic to U).This solves the main open problem of restricted dualities for minor closed classes and as an application it yields the bounded chromatic number of exact odd powers of any graph in an arbitrary proper minor closed class. We also generalize the decomposition theorem of DeVos et al. [M. DeVos, G. Ding, B. Oporowski, D.P. Sanders, B. Reed, P. Seymour, D. Vertigan, Excluding any graph as a minor allows a low tree-width 2-coloring, J. Combin. Theory Ser. B 91 (2004) 25-41].

Duality Theorems for Finite Structures (Characterising Gaps and Good Characterisations)

We provide a correspondence between the subjects of duality and density in classes of finite relational structures. The purpose of duality is to characterise the structures C that do not admit a homomorphism into a given target B by the existence of a homomorphism from a structure A into C. Density is the order-theoretic property of containing no covers (or “gaps”). We show that the covers in the skeleton of a category of finite relational models correspond naturally to certain instances of duality statements, and we characterise these covers.

The Ramsey property for graphs with forbidden complete subgraphs

Abstract The following theorem is proved: Let G be a finite graph with cl( G ) = m , where cl( G ) is the maximum size of a clique in G . Then for any integer r ≥ 1, there is a finite graph H , also with cl( H ) = m , such that if the edges of H are r -colored in any way, then H contains an induced subgraph G ′ isomorphic to G with all its edges the same color.

On classes of relations and graphs determined by subobjects and factorobjects

Abstract The classes of relations and graphs determined by subobjects and factorobjects are studied. We investigate whether such classes are closed under products, whether they are finitely generated by products and subobjects and whether a class can be described alternatively by subobjects and factorobjects. This is related to good characterizations.

The Mathematics of Paul Erdös I

VOLUME I.- Paul Erdos - Life and Work.- Paul Erdos Magic.- Part I Early Days.- Introduction.- Some of My Favorite Problems and Results.- 3 Encounters with Paul Erdos.- 4 Did Erdos Save Western Civilization?.- Integers Uniquely Represented by Certain Ternary Forms.- Did Erdos Save Western Civilization?.- Encounters with Paul Erdos.- On Cubic Graphs of Girth at Least Five.- Part II Number Theory.- Introduction.- Cross-disjoint Pairs of Clouds in the Interval Lattice.- Classical Results on Primitive and Recent Results on Cross-Primitive Sequences.- Dense Difference Sets and their Combinatorial Structure.- Integer Sets Containing No Solution to x+y=3z.- On Primes Recognizable in Deterministic Polynomial Time.- Ballot Numbers, Alternating Products, and the Erdos-Heilbronn Conjecture.- On Landaus Function g(n).- On Divisibility Properties on Sequences of Integers.- On Additive Representation Functions.- Arithmetical Properties of Polynomials.- Some Methods of Erdos Applied to Finite Arithmetic Progressions.- Sur La Non-Derivabilite de Fonctions Periodiques Associees a Certaines Formules Sommatoires.- 1105: First Steps in a Mysterious Quest.- Part III Randomness and Applications.- Introduction.- Games, Randomness, and Algorithms.- The Origins of the Theory of Random Graphs.- An Upper bound for a Communication Game Related to Time-space Tradeoffs.- How Abelian is a Finite Group?.- One Small Size Approximation Models.- The Erdos Existence Argument.- Part IV Geometry.- Introduction.- Extension of Functional Equations.- Remarks on Penrose Tilings.- Distances in Convex Polygons.- Unexpected Applications of Polynomials in Combinatorics.- The Number of Homothetic Subsets.- On Lipschitz Mappings Onto a Square.- A Remark on Transversal Numbers.- In Praise of the Gram Matrix.- On Mutually Avoiding Sets.- Bibliography.

On the maximum average degree and the oriented chromatic number of a graph

Abstract The oriented chromatic number o( H ) of an oriented graph H is defined as the minimum order of an oriented graph H ′ such that H has a homomorphism to H ′. The oriented chromatic number o( G ) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In this paper we study the links between o( G ) and mad( G ) defined as the maximum average degree of the subgraphs of G.

Survey: Colouring, constraint satisfaction, and complexity

Constraint satisfaction problems have enjoyed much attention since the early seventies, and in the last decade have become also a focus of attention amongst theoreticians. Graph colourings are a special class of constraint satisfaction problems; they offer a microcosm of many of the considerations that occur in constraint satisfaction. From the point of view of theory, they are well known to exhibit a dichotomy of complexity - the k-colouring problem is polynomial-time solvable when k@?2, and NP-complete when k>=3. Similar dichotomy has been proved for the class of graph homomorphism problems, which are intermediate problems between graph colouring and constraint satisfaction. However, for general constraint satisfaction problems, dichotomy has only been conjectured. Although the conjecture remains unproven to this day, it has been driving much of the theoretical research on constraint satisfaction problems, which combines methods of logic, universal algebra, analysis, and combinatorics. Currently, this is a very active area of research, and it is our goal here to present some of the recent developments, updating some of the information in existing books and surveys, while focusing on both the mathematical and the computational aspects of the theory. Given the level of activity, we are only able to survey a fraction of the new work, with emphasis on our own areas of interest.