Hedetniemi's Conjecture Research Articles

The homomorphism and coloring of the direct product of signed graphs

Hedetniemi's conjecture, a significant problem in graph theory, was disproved by Shitov in 2019. This breakthrough stimulates us to consider the signed graphs version of Hedetniemi's conjecture, where signed graphs are graphs with an assignment of {±1} to the edges. In this paper, we provide counterexamples to the signed analog of Hedetniemi's conjecture. Nevertheless, we show that the chromatic number of the product is determined by its factors in some instances, i.e., if Γ→Θ and (Θ,π) is balanced, then χ((Γ,σ)×(Θ,π))=χ((Γ,σ)). As a byproduct, we investigate the homomorphisms from a factor of (Γ,σ)×(Θ,π) to the other, say from (Γ,σ) to (Θ,π). We find that if such homomorphisms exist, then the chromatic number of (Γ,+) provides a lower bound for χ((Γ,σ)×(Θ,π)). Moreover, if Γ is a core, then the number of balanced subgraphs isomorphic to (Γ,+) in (Γ,σ)×(Θ,π) is equal to the number of such homomorphisms.

Counterexamples to Hedetniemi's conjecture and infinite Boolean lattices

We prove that for any $c \geq 5$, there exists an infinite family $(G_n )_{n\in \mathbb{N}}$ of graphs such that $\chi(G_n) > c$ for all $n\in \mathbb{N}$ and $\chi(G_m \times G_n) \leq c$ for all $m \neq n$. These counterexamples to Hedetniemi's conjecture show that the Boolean lattice of exponential graphs with $K_c$ as a base is infinite for $c \geq 5$.

Hedetniemi's conjecture from the topological viewpoint

This paper is devoted to studying a topological version of the famous Hedetniemi conjecture which says: The Z/2-index of the Cartesian product of two Z/2-spaces is equal to the minimum of their Z/2-indexes. We fully confirm the version of this conjecture for the homological index via establishing a stronger formula for the homological index of the join of Z/2-spaces. Moreover, we confirm the original conjecture for the case when one of the factors is an n-sphere. In particular, we answer a question about computing the index of some non-trivial products, raised by Marcin Wrochna.There is a generalization of Hedetniemi's conjecture for hypergraphs by Xuding Zhu. The surprising counterexample of the Hedetniemi conjecture which has been found very recently by Yaroslav Shitov shows that Hedetniemi's conjecture, and hence its generalization, only can be valid under additional assumptions. To approach Zhu's conjecture, we establish a topological lower bound for the chromatic number of the categorical product of hypergraphs. Consequently, we enrich the family of known hypergraphs satisfying Zhu's conjecture. In particular, this leads to a new proof for the fact that Zhu's conjecture is valid for the usual Kneser r-hypergraphs which has been established recently by Hossein Hajiabolhassan and Frédéric Meunier as a first non-trivial example of hypergraphs satisfying Zhu's conjecture.

Relatively small counterexamples to Hedetniemi's conjecture

Hedetniemi conjectured in 1966 that χ(G×H)=min⁡{χ(G),χ(H)} for all graphs G and H. Here G×H is the graph with vertex set V(G)×V(H) defined by putting (x,y) and (x′,y′) adjacent if and only if xx′∈E(G) and yy′∈E(H). This conjecture received a lot of attention in the past half century. Recently, Shitov refuted this conjecture. Let p be the minimum number of vertices in a graph of odd girth 7 and fractional chromatic number greater than 3+4/(p−1). Shitov's proof shows that Hedetniemi's conjecture fails for some graphs with chromatic number about p33p. In this paper, we show that the conjecture fails already for some graphs G and H with chromatic number 3⌈p+12⌉ and with p⌈(p−1)/2⌉ and 3⌈p+12⌉(p+1)−p vertices, respectively. The currently known upper bound for p is 83. Thus Hedetniemi's conjecture fails for some graphs G and H with chromatic number 126, and with 3,403 and 10,501 vertices, respectively.

Hedetniemi's conjecture is asymptotically false

Extending a recent breakthrough of Shitov, we prove that the chromatic number of the tensor product of two graphs can be a constant factor smaller than the minimum chromatic number of the two graphs. More precisely, we prove that there exists an absolute constant δ>0 such that for all c sufficiently large, there exist graphs G and H with chromatic number at least (1+δ)c for which χ(G×H)≤c.

Chromatic numbers, Sabidussi's Theorem and Hedetniemi's conjecture for non-commutative graphs

Non-commutative graph theory is an operator space generalization of graph theory. Well known graph parameters such as the independence number and Lovász theta function were first generalized to this setting by Duan, Severini, and Winter [1]. In [10], Stahlke introduces a generalization of the chromatic number of a graph and obtains a Lovász sandwich inequality for certain operator spaces.We introduce two new generalizations of the chromatic number to non-commutative graphs and provide an upper bound on the parameter of Stahlke. We provide a generalization of the graph complement and show the chromatic number of the orthogonal complement of a non-commutative graph is bounded below by its theta number. We also provide a generalization of both Sabidussi's Theorem and Hedetniemi's conjecture to non-commutative graphs.

On inverse powers of graphs and topological implications of Hedetniemi's conjecture

We consider a natural graph operation Ωk that is a certain inverse (formally: the right adjoint) to taking the k-th power of a graph. We show that it preserves the topology (the Z2-homotopy type) of the box complex, a basic tool in applications of topology in combinatorics. Moreover, we prove that the box complex of a graph G admits a Z2-map (an equivariant, continuous map) to the box complex of a graph H if and only if the graph Ωk(G) admits a homomorphism to H, for high enough k.This allows to show that if Hedetniemi's conjecture on the chromatic number of graph products is true, then the following analogous conjecture in topology is also true: If n∈N and X,Y are Z2-spaces (finite Z2-simplicial complexes) such that X×Y admits a Z2-map to the n-dimensional sphere, then X or Y itself admits such a map. We discuss this and other implications, arguing the importance of the topological conjecture.

Hedetniemi's Conjecture and Strongly Multiplicative Graphs

A graph $K$ is multiplicative if a homomorphism from any product $G \times H$ to $K$ implies a homomorphism from $G$ or from $H$. Hedetniemi's conjecture stated that all cliques are multiplicative. In an attempt to explore the boundaries of current methods, we investigate strongly multiplicative graphs, which we define as graphs $K$ 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 nonbipartite, $H$ is a connected graph with a vertex $h$, and there is a homomorphism $\phi\!: G \times H \rightarrow K$ such that $ \phi(-,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 one 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.

Hedetniemi's conjecture for uncountable graphs

It is proved that in Gödel's constructible universe, for every infinite successor cardinal \kappa , there exist graphs \mathcal G and \mathcal H of size and chromatic number \kappa , for which the product graph \mathcal G \times \mathcal H is countably chromatic. In particular, this provides an affirmative answer to a question of Hajnal from 1985.

Square-free graphs are multiplicative

A graph K is square-free if it contains no four-cycle as a subgraph (i.e., for every quadruple of vertices, if v0v1,v1v2,v2v3,v3v0∈E(K), then v0=v2 or v1=v3). A graph K is multiplicative if G×H→K implies G→K or H→K, for all graphs G, H. Here G×H is the tensor (or categorical) graph product and G→K denotes the existence of a graph homomorphism from G to K. Hedetniemi's conjecture, which states that χ(G×H)=min⁡(χ(G),χ(H)), is equivalent to the statement that all cliques Kn are multiplicative. However, the only non-trivial graphs known to be multiplicative are K3, odd cycles, and still more generally, circular cliques Kp/q with 2≤pq<4. We make no progress for cliques, but show that all square-free graphs are multiplicative. In particular, this gives the first multiplicative graphs of chromatic number higher than 4. Generalizing, in terms of the box complex, the topological insight behind existing proofs for odd cycles, we also give a different proof for circular cliques with 2≤pq<4.

Hedetniemi's conjecture for Kneser hypergraphs

One of the most famous conjectures in graph theory is Hedetniemi's conjecture stating that the chromatic number of the categorical product of graphs is the minimum of their chromatic numbers. Using a suitable extension of the definition of the categorical product, Zhu proposed in 1992 a similar conjecture for hypergraphs. We prove that Zhu's conjecture is true for the usual Kneser hypergraphs of same rank. It provides to the best of our knowledge the first non-trivial and explicit family of hypergraphs with rank larger than two satisfying this conjecture (the rank two case being Hedetniemi's conjecture). We actually prove a more general result providing a lower bound on the chromatic number of the categorical product of any Kneser hypergraphs as soon as they all have same rank. We derive from it new families of graphs satisfying Hedetniemi's conjecture. The proof of the lower bound relies on the Zp-Tucker lemma.

Sabidussi versus Hedetniemi for three variations of the chromatic number

We investigate vector chromatic number (źvec), Lovasz V-function of the complement $$(\bar \vartheta )$$, and quantum chromatic number (źq) from the perspective of graph homomorphisms. We prove an analog of Sabidussi's theorem for each of these parameters, i.e., that for each of the parameters, the value on the Cartesian product of graphs is equal to the maximum of the values on the factors. Interestingly, as a consequence of this result for $$\bar \vartheta$$, we obtain analog of Hedetniemi's conjecture, i.e., that the value of $$\bar \vartheta$$ on the categorical product of graphs is equal to the minimum of its values on the factors. We conjecture that the analogous results hold for vector and quantum chromatic number, and we prove that this is the case for some special classes of graphs.

Hedetniemi's Conjecture and Fiber Products of Graphs

We prove that for n ≥ 4, a fiber product of n-chromatic graphs over n-colourings can have chromatic number strictly less than n. This refutes a conjecture of A. Khelladi, Sur une opération binaire de graphes, Communication to the 32d Week of Science, Damas (Syria, 7-13 November 1992).

Coloring Fiber Product of Graphs

The fiber product of graphs over homomorphims, a notion introduced by P. Hell (1972) is used here as a new approach to S. Hedetniemi's conjecture (1966) since it's a subgraph of the cross product. We consider here only the fiber product over colorings which are special cases of homomorphisms of graphs. We show that Khelladi's conjecture (1991): “The fiber product over colorings of two n-chromatic graphs is also an n-chromatic graph” implying trivially Hedetniemi's is true for n ≥ 3. Moreover, we propose an equivalent statement of Khelladi's conjecture using the class of graph colourings of a graph defined by El-Zahar and Sauer.

Generic automorphisms and graph coloring

The question of whether or not a given countable arithmetically saturated model of Peano Arithmetic has a generic automorphism is shown to be very closely connected to Hedetniemi's well-known conjecture on the chromatic number of products of graphs.

The level of nonmultiplicativity of graphs

We introduce a parameter called the level of nonmultiplicativity of a graph, which is related to Hedetniemi's conjecture. We show that this parameter is equal to the number of factors in a factorization of the graph into a product of multiplicative graphs. Apart from the known multiplicative graphs, no graph is known to have a finite level of nonmultiplicativity. We show that the countably infinite complete graph K ℵ 0 has an infinite level of nonmultiplicativity and that there exist Kneser graphs with arbitrarily high levels of nonmultiplicativity.

On Hedetniemi's conjecture and the colour template scheme

Hedetniemi's conjecture states that the chromatic number of a product of graphs is the minimum of the chromatic numbers of the factors. El-Zahar and Sauer (Combinatorica 5 (1985) 121) have shown that the chromatic number of a product of 4-chromatic graphs is 4. In this paper, we investigate to which extent their methods can be adapted to the case of 5-chromatic graphs. Our main result is that if G and H are connected graphs of chromatic number at least 5 that both contain odd wheels as subgraphs, then their product also has chromatic number at least 5. However, we also provide counterexamples to the El-Zahar and Sauer conjecture on the structure of n-chromatic subgraphs of products of n-chromatic graphs. This implies that the method of El-Zahar and Sauer will not be sufficient to prove all cases of Hedetniemi's conjecture for chromatic numbers larger than 4.

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

Lattices arising in categorial investigations of Hedetniemi's conjecture

We discuss Hedetniemi's conjecture in the context of categories of relational structures under homomorphisms. In this language Hedetniemi's conjecture says that if there are no homomorphisms from the graphs G and H to the complete graph on n vertices then there is no homomorphism from G × H to the complete graph. If an object in some category has just this property then it is called multiplicative. The skeleton of a category of relational structures under homomorphisms forms a distributive lattice which has for each of the objects K of the category a pseudocomplementation. The image of the distributive lattice under such a pseudo-complementation is a Boolean lattice with the same meet as the distributive lattice and the structure K is multiplicative if and only if this Boolean lattice consists of at most two elements. We will exploit those general ideas to gain some understanding of the situation in the case of graphs and solve completely the Hedetniemi-type problem in the case of relational structures over a unary language.

An approach to hedetniemi's conjecture

AbstractFor a fixed integer n ϵ ω, a graph G of chromatic number greater than n is called persistent if for all n + 1‐chromatic graphs H, the products G × H are n + 1‐chromatic graphs. Wheter all graphs of chromatic number greater than n are persistent is a long‐standing open problem due to Hedetniemi. We call a graph G strongly persistent if G is persistent and the Hajos sum of G with any other persistent graph H is still persistent. This paper extends the class of known persistent graphs by proving the following results: If G is constructed from copies of Kn+1 by Hajos sums, adding vertices and edges and at most one contraction of nonadjacent vertices, then G is strongly persistent. © 1929 John Wiley &amp; Sons, Inc.