Hedetniemi's conjecture

Example of Hedetniemi's conjecture: the tensor product of C5 and C3 (on the left) produces a graph that contains a cycle with length 15 (on the right) so: the resulting graph requires 3 colors.

In graph theory, Hedetniemi's conjecture, formulated by Stephen T. Hedetniemi in 1966, concerns the connection between graph coloring and the tensor product of graphs. This conjecture states that

\chi (G\times H)=\min\{\chi (G),\chi (H)\}.

Here $\chi (G)$ denotes the chromatic number of an undirected finite graph $G$ .

The inequality χ(G × H) ≤ min {χ(G), χ(H)} is easy: if G is k-colored, one can k-color G × H by using the same coloring for each copy of G in the product; symmetrically if H is k-colored. Thus, Hedetniemi's conjecture amounts to the assertion that tensor products cannot be colored with an unexpectedly small number of colors.

A counterexample to the conjecture was discovered by Yaroslav Shitov (2019) (see Kalai 2019), thus disproving the conjecture in general.

Known cases

Any graph with a nonempty set of edges requires at least two colors; if G and H are not 1-colorable, that is, they both contain an edge, then their product also contains an edge, and is hence not 1-colorable either. In particular, the conjecture is true when G or H is a bipartite graph, since then its chromatic number is either 1 or 2.

Similarly, if two graphs G and H are not 2-colorable, that is, not bipartite, then both contain a cycle of odd length. Since the product of two odd cycle graphs contains an odd cycle, the product G × H is not 2-colorable either. In other words, if G × H is 2-colorable, then at least one of G and H must be 2-colorable as well.

The next case was proved long after the conjecture's statement, by El-Zahar & Sauer (1985): if the product G × H is 3-colorable, then one of G or H must also be 3-colorable. In particular, the conjecture is true whenever G or H is 4-colorable (since then the inequality χ(G × H) ≤ min {χ(G), χ(H)} can only be strict when G × H is 3-colorable). In the remaining cases, both graphs in the tensor product are at least 5-chromatic and progress has only been made for very restricted situations.

Weak Hedetniemi Conjecture

The following function (known as the Poljak-Rödl function) measures how low the chromatic number of products of n-chromatic graphs can be.

f(n)=\min\{\chi (G\times H)\colon \chi (G)=\chi (H)=n\}

Hedetniemi's conjecture is then equivalent to saying that $f (n) = n$ . The Weak Hedetniemi Conjecture instead states merely that the function f(n) is unbounded. In other words, if the tensor product of two graphs can be colored with few colors, this should imply some bound on the chromatic number of one of the factors.

The main result of (Poljak & Rödl 1981), independently improved by Poljak, James H. Schmerl, and Zhu, states that if the function f(n) is bounded, then it is bounded by at most 9. Thus a proof of Hedetniemi's conjecture for 10-chromatic graphs would already imply the Weak Hedetniemi Conjecture for all graphs.

Multiplicative graphs

The conjecture is studied in the more general context of graph homomorphisms, especially because of interesting relations to the category of graphs (with graphs as objects and homomorphisms as arrows). For any fixed graph K, one considers graphs G that admit a homomorphism to K, written G → K. These are also called K-colorable graphs. This generalizes the usual notion of graph coloring, since it follows from definitions that a k-coloring is the same as a K_k-coloring (a homomorphism into the complete graph on k vertices).

A graph K is called multiplicative if for any graphs G, H, the fact that G × H → K holds implies that G → K or H → K holds. As with classical colorings, the reverse implication always holds: if G (or H, symmetrically) is K-colorable, then G × H is easily K-colored by using the same values independently of H. Hedetniemi's conjecture is then equivalent to the statement that each complete graph is multiplicative.

The above known cases are equivalent to saying that K₁, K₂, and K₃ are multiplicative. The case of K₄ is widely open. On the other hand, the proof of El-Zahar & Sauer (1985) has been generalized by Häggkvist et al. (1988) to show that all cycle graphs are multiplicative. Later, Tardif (2005) proved more generally that all circular cliques K_n/k with n/k < 4 are multiplicative. In terms of the circular chromatic number χ_c, this means that if $χ c (G \times H) < 4$ , then $χ c (G \times H) = min {χ c (G), χ c (G)}$ . Wrochna (2017) has shown that square-free graphs are multiplicative.

Examples of non-multiplicative graphs can be constructed from two graphs G and H that are not comparable in the homomorphism order (that is, neither G→H nor H→G holds). In this case, letting K=G×H, we trivially have G×H→K, but neither G nor H can admit a homomorphism into K, since composed with the projection K→H or K→G it would give a contradiction.

Exponential graph

Since the tensor product of graphs is the category-theoretic product in the category of graphs (with graphs as objects and homomorphisms as arrows), the conjecture can be rephrased in terms of the following construction on graphs K and G. The exponential graph $K G$ is the graph with all functions $V(G) \to V(K)$ as vertices (not only homomorphisms) and two functions f,g adjacent when

f(v) is adjacent to g(v') in K, for all adjacent vertices v,v ' of G.

In particular, there is a loop at a function f (it is adjacent to itself) if and only if the function gives a homomorphism from G to K. Seen differently, there is an edge between f and g whenever the two functions define a homomorphism from G × K₂ (the bipartite double cover of G) to K.

The exponential graph is the exponential object in the category of graphs. This means homomorphisms from G × H to a graph K correspond to homomorphisms from H to K^G. Moreover, there is a homomorphism $eval : G \times K G \to K$ given by $eval(v, f) = f (v)$ . These properties allow to conclude that the multiplicativity of K is equivalent (El-Zahar & Sauer 1985) to the statement:

either G or K^G is K-colorable, for every graph G.

In other words, Hedetniemi's conjecture can be seen as a statement on exponential graphs: for every integer k, the graph K_k^G is either k-colorable, or it contains a loop (meaning G is k-colorable). One can also see the homomorphisms $eval : G \times K k G \to K k$ as the hardest instances of Hedetniemi's conjecture: if the product G × H was a counterexample, then G × K_k^G would also be a counterexample.

Generalizations

Generalized to directed graphs, the conjecture has simple counterexamples, as observed by Poljak & Rödl (1981). Here, the chromatic number of a directed graph is just the chromatic number of the underlying graph, but the tensor product has exactly half the number of edges (for directed edges g→g' in G and h→h' in H, the tensor product G × H has only one edge, from (g,h) to (g',h'), while the product of the underlying undirected graphs would have an edge between (g,h') and (g',h) as well). However, the Weak Hedetniemi Conjecture turns out to be equivalent in the directed and undirected settings (Tardif & Wehlau 2006).

The problem cannot be generalized to infinite graphs: Hajnal (1985) gave an example of two infinite graphs, each requiring an uncountable number of colors, such that their product can be colored with only countably many colors. Rinot (2013) proved that in the constructible universe, for every infinite cardinal $\kappa$ , there exist a pair of graphs of chromatic number greater than $\kappa$ , such that their product can still be colored with only countably many colors.

References

Primary sources

Duffus, D.; Sands, B.; Woodrow, R. E. (1985), "On the chromatic number of the product of graphs", Journal of Graph Theory, 9 (4): 487–495, doi:10.1002/jgt.3190090409, MR 0890239.
El-Zahar, M.; Sauer, N. (1985), "The chromatic number of the product of two 4-chromatic graphs is 4", Combinatorica, 5 (2): 121–126, doi:10.1007/BF02579374, MR 0815577, S2CID 7659747.
Häggkvist, R.; Hell, P.; Miller, D. J.; Neumann Lara, V. (1988), "On multiplicative graphs and the product conjecture", Combinatorica, 8 (1): 63–74, doi:10.1007/BF02122553, hdl:1828/1589, MR 0951994, S2CID 39731578.
Hajnal, A. (1985), "The chromatic number of the product of two ℵ₁ chromatic graphs can be countable", Combinatorica, 5 (2): 137–140, doi:10.1007/BF02579376, MR 0815579, S2CID 27087122.
Hedetniemi, S. (1966), Homomorphisms of graphs and automata, Technical Report 03105-44-T, University of Michigan.
Poljak, S.; Rödl, V. (1981), "On the arc-chromatic number of a digraph", Journal of Combinatorial Theory, Series B, 31 (2): 190–198, doi:10.1016/S0095-8956(81)80024-X.
Rinot, A. (2013), Hedetniemi's conjecture for uncountable graphs, arXiv:1307.6841, Bibcode:2013arXiv1307.6841R.
Sabidussi, G. (1957), "Graphs with given group and given graph-theoretical properties", Canadian Journal of Mathematics, 9: 515–525, doi:10.4153/CJM-1957-060-7, MR 0094810, S2CID 124514137.
Shitov, Yaroslav (May 2019), Counterexamples to Hedetniemi's conjecture, arXiv:1905.02167.
Tardif, C. (2005), "Multiplicative graphs and semi-lattice endomorphisms in the category of graphs", Journal of Combinatorial Theory, Series B, 95 (2): 338–345, doi:10.1016/j.jctb.2005.06.002.
Tardif, C.; Wehlau, D. (2006), "Chromatic numbers of products of graphs: The directed and undirected versions of the Poljak-Rödl function", Journal of Graph Theory, 51 (1): 33–36, doi:10.1002/jgt.20117, S2CID 17489968.
Wrochna, M. (2017), "Square-free graphs are multiplicative", Journal of Combinatorial Theory, Series B, 122: 479–507, arXiv:1601.04551, doi:10.1016/j.jctb.2016.07.007, S2CID 205930734.

Surveys and secondary sources

Imrich, Wilfried; Klavžar, Sandi (2000), Product Graphs: Structure and Recognition, Wiley, ISBN 0-471-37039-8.
Kalai, Gil (May 10, 2019), "A sensation in the morning news – Yaroslav Shitov: Counterexamples to Hedetniemi's conjecture", Combinatorics and More.
Klavžar, Sandi (1996), "Coloring graph products: a survey", Discrete Mathematics, 155 (1–3): 135–145, doi:10.1016/0012-365X(94)00377-U, MR 1401366.
Sauer, N. (2001), "Hedetniemi's conjecture: a survey", Discrete Mathematics, 229 (1–3): 261–292, doi:10.1016/S0012-365X(00)00213-2, MR 1815610.
Tardif, Claude (2008), "Hedetniemi's conjecture, 40 years later" (PDF), Graph Theory Notes of New York, 54: 46–57, MR 2445666.
Zhu, Xuding (1998), "A survey on Hedetniemi's conjecture", Taiwanese J. Math., 2 (1): 1–24, doi:10.11650/twjm/1500406890, MR 1609464.

External links

A breakthrough in graph theory, Numberphile

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