Hedetniemi conjecture

For two graphs $A$ and $B$ (cf. Graph), their categorial product $A\times B$ is the graph with vertex set $V(A\times B)=V(A)\times V(B)$ while an edge $(a_0,b_0)$ is adjacent to an edge $(a_1,b_1)$ if and only if $a_0$ is adjacent to $a_1$ and $b_0$ is adjacent to $b_1$. If the graph $A$ has a good $n$-colouring $\gamma$ (cf. Graph colouring), then $A\times B$ has a good $n$-colouring $\gamma^*$, given by $\gamma^*(a,b)=\gamma(a)$. As this statement also holds for good $n$-colourings of $B$, it follows that the chromatic number $\chi(A\times B)\leq\min\{\chi(A),\chi(B)\}$. Hedetniemi's conjecture, [a1], is that for all finite graphs $A$ and $B$:

$$\chi(A\times B)=\min\{\chi(A),\chi(B)\}.$$

The corresponding statement for infinite graphs is not true [a2], but little is known about the relationship between the cardinalities of the chromatic numbers of $A$ and $B$ and the chromatic number of $A\times B$.

It is obvious that the chromatic number of the product of two $2$-chromatic graphs is $2$. L. Lovász [a3] showed that the chromatic number of the product of two $3$-chromatic graphs is $3$ and in [a4] it is proven that the chromatic number of the product of two $4$-chromatic graphs is $4$. It is not known whether the chromatic number of the product of two $n$-chromatic graphs is always $n$ for any $n\geq5$. The lack of understanding of this problem is even more embarrassing than that.

Let $f(n)$ be the largest number such that whenever $A$ and $B$ are two $n$-chromatic graphs, then the chromatic number of $A\times B$ is larger than or equal to $f(n)$. Clearly, if $n\geq4$, then $4\leq f(n)\leq n$ and Hedetniemi's conjecture says that $f(n)=n$. About $f(n)$ it is only known that either $f(n)\leq9$ for all $n$ or $f(n)$ is unbounded, [a5] and [a6].

Write $A\mapsto B$ if there is a homomorphism from the graph $A$ to the graph $B$ (a function from $A$ to $B$ which preserves the edges). Write $A\not\mapsto B$ if there is no homomorphism from $A$ to $B$. A graph $M$ is called multiplicative, [a7], [a8], if $A\not\mapsto M$ and $B\not\mapsto M$ imply $A\times B\not\mapsto M$. Note that a graph $A$ has a good $n$-colouring if and only if $A\mapsto K_n$. Hence Hedetniemi's conjecture is equivalent to asking whether the complete graph $K_n$ (cf. also Graph) is multiplicative.

Two graphs $A$ and $B$ are equivalent, $A\sim B$, if $A\mapsto B$ and $B\mapsto A$. It is not difficult to check that the equivalence classes of finite graphs form a distributive lattice under the order relation $\mapsto$ and that a graph is multiplicative if and only if it is meet-irreducible in this lattice. Once this general setting of the problem has been realized, it is not difficult to see that Hedetniemi's conjecture has a natural generalization, [a8], to the problem of finding the meet-irreducible elements in the distributive lattices of various types of relational structures. This line of investigation leads to a setting in topos theory and, to a certain extent, in logic, [a9].

Of course this introduces the even more general task of trying to understand the distributive lattice of, say, all finite relational structures of a given type and, in connection with this, the problem of deciding whether there is a homomorphism from a given relational structure to another one of the same type. In this context one of the more interesting types of relational structures are finite directed graphs, [a10]. Multiplicative partially ordered sets have been investigated in [a11].

For graphs, Hedetniemi's conjecture has been confirmed for various special cases and several interesting observations have been made. See [a12], [a13], [a14]. For a general discussion of graph products, see [a15].

References

[a1]	S. Hedetniemi, "Homomorphisms of graphs and automata" Univ. Michigan Technical Report , 03105–44–T (1966)
[a2]	A. Hajnal, "The chromatic number of the product of two $KAPAJA_1$-chromatic graphs can be countable" Combinatorica , 5 (1985) pp. 137–140
[a3]	L. Lovász, "Operations with structures" Acta Math. Acad. Sci. Hung. (1967) pp. 321–328
[a4]	M.H. El-Zahar, N.W. Sauer, "The chromatic number of the product of two $4$-chromatic graphs is $4$" Combinatorica , 5 : 2 (1985) pp. 121–126
[a5]	S. Poljak, "Coloring digraphs by iterated antichains" Comment. Math. Univ. Carolin. , 32 : 2 (1991) pp. 209–212
[a6]	S. Poljak, V. Rödl, "On the arc-chromatic number of a digraph" JCT B , 31 (1981) pp. 190–198
[a7]	D.J. Miller, "The categorical product of graphs" Canad. J. Math. , 20 (1968) pp. 1511–1521
[a8]	R. Haggkvist, P. Hell, D.J. Miller, V.N. Lara, "On multiplicative graphs and the product conjecture" Combinatorica , 8 : 1 (1988) pp. 63–74
[a9]	D. Duffus, N.W. Sauer, "Lattices arising in categorial investigations of Hedetniemi's conjecture" Discrete Math. , 153 (1996)
[a10]	Hell, Pavol; Zhou, Huishan; Zhu, Xuding "Homomorphisms to oriented cycles" Combinatorica 13, No.4 (1993) 421-433 Zbl 0794.05037
[a11]	N.W. Sauer, X. Zhu, "Multiplicative posets" Order , 8 (1992) pp. 349–358
[a12]	D. Duffus, B. Sands, R. Woodrow, "On the chromatic number of the product of graphs" J. Graph Th. , 9 (1985) pp. 487–495
[a13]	N.W. Sauer, X. Zhu, "An approach to Hedetniemi's conjecture" J. Graph Th. , 16 : 5 (1992) pp. 423–436
[a14]	G. Sabidussi, "Graph multiplication" Math. Z. , 72 (1960) pp. 446–457
[a15]	Nowakowski, Richard J.; Rall, Douglas F. "Associative graph products and their independence, domination and coloring numbers" Discuss. Math., Graph Theory 16, No.1 (1996) 53-79 {{ZBL|0865.05071}