Claude Tardif

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.

Graph homomorphisms: structure and symmetry

This paper is the first part of an introduction to the subject of graph homomorphism in the mixed form of a course and a survey. We give the basic definitions, examples and uses of graph homomorphisms and mention some results that consider the structure and some parameters of the graphs involved. We discuss vertex-transitive graphs and Cayley graphs and their rather fundamental role in some aspects of graph homomorphisms. Graph colourings are then explored as homomorphisms, followed by a discussion of various graph products.

Projectivity and independent sets in powers of graphs

We investigate the relationship between projectivity and the structure of maximal independent sets in powers of circular graphs, Kneser graphs and truncated simplices. 2002 Wiley Periodicals, Inc. J Graph Theory 40: 162– 171, 2002

Prefibers and the cartesian product of metric spaces

Abstract The properties of certain sets called prefibers in a metric space are used to show that the algebraic properties of the cartesian product of graphs generalize to metric spaces.

Multiplicative graphs and semi-lattice endomorphisms in the category of graphs

A graph K is called multiplicative if whenever a categorical product of two graphs admits a homomorphism to K, then one of the factors also admits a homomorphism to K. We prove that all circular graphs Kk/d such that k/d < 4 are multiplicative. This is done using semi-lattice endomorphism in (the skeleton of) the category of graphs to prove the multiplicativity of some graphs using the known multiplicativity of the odd cycles.

On Normal Cayley Graphs and Hom-idempotent Graphs

A graphGis said to behom-idempotentif there is a homomorphism fromG2toG, andweakly hom-idempotentif for somen ? 1 there is a homomorphism fromGn+ 1toGn. We characterize both classes of graphs in terms of a special class of Cayley graphs callednormal Cayley graphs. This allows us to construct, for any integern, a Cayley graphGsuch thatGn+1?Gn?Gn?1, answering a question of Hahn, Hell and Poljak 8. Also, we show that the Kneser graphs are not weakly hom-idempotent, generalizing a result of Albertson and Collins 1 for the Petersen graph.

A Characterisation of First-Order Constraint Satisfaction Problems

We characterise finite relational core structures admitting finitely many obstructions, in terms of special near unanimity functions, and in terms of dismantling properties of their square. As a consequence, we show that it is decidable to determine whether a constraint satisfaction problem is first-order definable: we show the general problem to be NP-complete, and give a polynomial-time algorithm in the case of cores

Hedetniemi's Conjecture and the Retracts of a Product of Graphs

AbstractWe show that every core graph with a primitive automorphismgroup has the property that whenever it is a retract of a product ofconnected graphs, it is a retract of a factor. The example of Knesergraphs shows that the hypothesis that the factors are connected isessential. In the case of complete graphs, our result has already beenshown in [4, 17], and it is an instance where Hedetniemi’s conjectureis known to hold. In fact, our work is motivated by a reinterpretationof Hedetniemi’s conjecture in terms of products and retracts. 1 Introduction One of the outstanding problems in graph theory is a formula concerning thechromatic number of a product of graphs:Conjecture 1.1 (Hedetniemi [9]) χ(G×H) = min{χ(G),χ(H)}.Here, G× His the product of Gand H, defined byV(G× H) = V(G)×V(H)E(G× H) = {[(u 1 ,u 2 ),(v 1 ,v 2 )] : [u 1 ,v 1 ] ∈ E(G) and [u 2 ,v 2 ] ∈ E(H)}.A colouring of G× H can be derived from a colouring of any of its fac-tors, hence χ(G× H) ≤ min{χ(G),χ(H)}. The inherent difficulty of Con-jecture 1.1 lies in finding a lower bound for χ(G× H). It is known that1

On maximal finite antichains in the homomorphism order of directed graphs

We show that the pairs {T, DT } where T is a tree and DT its dual are the only maximal antichains of size 2 in the category of directed graphs endowed with its natural homomorphism ordering.

Graph products and the chromatic difference sequence of vertex-transitive graphs

Abstract We give examples of vertex-transitive graphs with non-monotonic chromatic difference sequences, disproving a conjecture of Albertson and Collins (1985) on the monotonicity of the chromatic difference sequence of vertex-transitive graphs, and answering a question of Zhou (1991) on the achievability of circulants.