Searcharxiv⌕ Search

arXiv subjects

Claude Tardif

Publications and source records attributed to Claude Tardif.

18 recordsLinked to original sources
arXiv2025

Constraint satisfaction problems, compactness and non-measurable sets

A finite relational structure A is called compact if for any infinite relational structure B of the same type, the existence of a homomorphism from B to A is equivalent to the existence of homomorphisms from all finite substructures of B to A. We show that if A has width one, then the compactness of A can be proved in the axiom system of Zermelo and Fraenkel, but otherwise, the compactness of A implies the existence of non-measurable sets in 3-space.

math.LO↗
arXiv2018

Hedetniemi's conjecture and strongly multiplicative graphs

A graph K is multiplicative if a homomorphism from any product G x H to K implies a homomorphism from G or from H. Hedetniemi's conjecture states that all cliques are multiplicative. In an attempt to explore the boundaries of current methods, we investigate strongly multiplicative graphs, which we define as K such that for any connected graphs G,H with odd cycles C,C', a homomorphism from $(G \times C') \cup (C \times H) \subseteq G \times H$ to K implies a homomorphism from G or H. Strong multiplicativity of K also implies the following property, which may be of independent interest: if G is non-bipartite, H is a connected graph with a vertex h, and there is a homomorphism $ϕ\colon G \times H \to K$ such that $ϕ(-,h)$ is constant, then H admits a homomorphism to K. All graphs currently known to be multiplicative are strongly multiplicative. We revisit the proofs in a different view based on covering graphs and replace fragments with more combinatorial arguments. This allows us to find new (strongly) multiplicative graphs: all graphs in which every edge is in at most square, and the third power of any graph of girth >12. Though more graphs are amenable to our methods, they still make no progress for the case of cliques. Instead we hope to understand their limits, perhaps hinting at ways to further extend them.

math.CO↗
arXiv2017

Logical compactness and constraint satisfaction problems

We investigate a correspondence between the complexity hierarchy of constraint satisfaction problems and a hierarchy of logical compactness hypotheses for finite relational structures. It seems that the harder a constraint satisfaction problem is, the stronger the corresponding compactness hypothesis is. At the top level, the NP-complete constraint satisfaction problems correspond to compactness hypotheses that are equivalent to the ultrafilter axiom in all the cases we have investigated. At the bottom level, the simplest constraint satisfaction problems correspond to compactness hypotheses that are readily provable from the axioms of Zermelo and Fraenkel.

math.LO↗
arXiv2016

Iterated Arc Graphs

The arc graph $δ(G)$ of a digraph $G$ is the digraph with the set of arcs of $G$ as vertex-set, where the arcs of $δ(G)$ join consecutive arcs of $G$. In 1981, Poljak and Rödl characterised the chromatic number of $δ(G)$ in terms of the chromatic number of $G$ when $G$ is symmetric (i.e., undirected). In contrast, directed graphs with equal chromatic numbers can have arc graphs with distinct chromatic numbers. Even though the arc graph of a symmetric graph is not symmetric, we show that the chromatic number of the iterated arc graph $δ^k(G)$ still only depends on the chromatic number of $G$ when $G$ is symmetric.

math.CO↗
arXiv2016

Topologically $4$-chromatic graphs and signatures of odd cycles

We investigate group-theoretic "signatures" of odd cycles of a graph, and their connections to topological obstructions to 3-colourability. In the case of signatures derived from free groups, we prove that the existence of an odd cycle with trivial signature is equivalent to having the coindex of the hom-complex at least 2 (which implies that the chromatic number is at least 4). In the case of signatures derived from elementary abelian 2-groups we prove that the existence of an odd cycle with trivial signature is a sufficient condition for having the index of the hom-complex at least 2 (which again implies that the chromatic number is at least 4).

math.CO↗
arXiv2016

Generalised Mycielski graphs and bounds on chromatic numbers

We prove that the coindex of the box complex $\mathrm{B}(H)$ of a graph $H$ can be measured by the generalised Mycielski graphs which admit a homomorphism to it. As a consequence, we exhibit for every graph $H$ a system of linear equations solvable in polynomial time, with the following properties: If the system has no solutions, then $\mathrm{coind}(\mathrm{B}(H)) + 2 \leq 3$; if the system has solutions, then $χ(H) \geq 4$. We generalise the method to other bounds on chromatic numbers using linear algebra.

math.CO↗
arXiv2015

Regular families of forests, antichains and duality pairs of relational structures

Homomorphism duality pairs play crucial role in the theory of relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper we characterize the infinite-finite antichain dualities and infinite-finite dualities with trees or forest on the left hand side. This work builds on our earlier papers that gave several examples of infinite-finite antichain duality pairs of directed graphs and a complete characterization for caterpillar dualities.

math.CO↗
arXiv2015

Homomorphisms of binary Cayley graphs

A binary Cayley graph is a Cayley graph based on a binary group. In 1982, Payan proved that any non-bipartite binary Cayley graph must contain a generalized Mycielski graph of an odd-cycle, implying that such a graph cannot have chromatic number 3. We strengthen this result first by proving that any non-bipartite binary Cayley graph must contain a projective cube as a subgraph. We further conjecture that any homo- morphism of a non-bipartite binary Cayley graph to a projective cube must be surjective and we prove some special case of this conjecture.

math.CO↗
arXiv2014

Digraph functors which admit both left and right adjoints

For our purposes, two functors Λ and Γ are said to be respectively left and right adjoints of each other if for any digraphs G and H, there exists a homomorphism of Λ(G) to H if and only if there exists a homomorphism of G to Γ(H). We investigate the right adjoints characterised by Pultr in [A. Pultr, The right adjoints into the categories of relational systems, in Reports of the Midwest Category Seminar, IV, volume 137 of Lecture Notes in Mathematics, pages 100-113, Berlin, 1970]. We find necessary conditions for these functors to admit right adjoints themselves. We give many examples where these necessary conditions are satisfied, and the right adjoint indeed exists. Finally, we discuss a connection between these right adjoints and homomorphism dualities.

math.CO↗
arXiv2013

Adjoint functors in graph theory

We survey some uses of adjoint functors in graph theory pertaining to colourings, complexity reductions, multiplicativity, circular colourings and tree duality. The exposition of these applications through adjoint functors unifies the presentation to some extent, and also raises interesting questions.

math.CO↗
arXiv2012

On infinite-finite duality pairs of directed graphs

The (A,D) duality pairs play crucial role in the theory of general relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite-finite case. Here we concentrate on directed graphs. We prove some elementary properties of the infinite-finite duality pairs, including lower and upper bounds on the size of D, and show that the elements of A must be equivalent to forests if A is an antichain. Then we construct instructive examples, where the elements of A are paths or trees. Note that the existence of infinite-finite antichain dualities was not previously known.

math.CO↗
arXiv2012

Caterpillar dualities and regular languages

We characterize obstruction sets in caterpillar dualities in terms of regular languages, and give a construction of the dual of a regular family of caterpillars. We show that these duals correspond to the constraint satisfaction problems definable by a monadic linear Datalog program with at most one EDB per rule.

math.CO↗
arXiv2010

Interleaved adjoints on directed graphs

For an integer k >= 1, the k-th interlacing adjoint of a digraph G is the digraph i_k(G) with vertex-set V(G)^k, and arcs ((u_1, ..., u_k), (v_1, ..., v_k)) such that (u_i,v_i) \in A(G) for i = 1, ..., k and (v_i, u_{i+1}) \in A(G) for i = 1, ..., k-1. For every k we derive upper and lower bounds for the chromatic number of i_k(G) in terms of that of G. In particular, we find tight bounds on the chromatic number of interlacing adjoints of transitive tournaments. We use this result in conjunction with categorial properties of adjoint functors to derive the following consequence. For every integer ell, there exists a directed path Q_{\ell} of algebraic length ell which admits homomorphisms into every directed graph of chromatic number at least 4. We discuss a possible impact of this approach on the multifactor version of the weak Hedetniemi conjecture.

math.CO↗
arXiv2009

Adjoint functors and tree duality

A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions.

math.CO↗
arXiv2007

A Characterisation of First-Order Constraint Satisfaction Problems

We describe simple algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions. 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. A slight modification of this algorithm provides, for first-order definable CSP's, a simple poly-time algorithm to produce a solution when one exists. As an application of our algebraic characterisation of first order CSP's, we describe a large family of L-complete CSP's.

cs.LO↗