View source for Hedetniemi conjecture

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For two graphs $A$ and $B$ (cf. [[Graph|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|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, [[#References|[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 [[#References|[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 [[#References|[a3]]] showed that the chromatic number of the product of two $3$-chromatic graphs is $3$ and in [[#References|[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, [[#References|[a5]]] and [[#References|[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, [[#References|[a7]]], [[#References|[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|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|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, [[#References|[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|topos]] theory and, to a certain extent, in logic, [[#References|[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, [[#References|[a10]]]. Multiplicative partially ordered sets have been investigated in [[#References|[a11]]].

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Hedetniemi,   "Homomorphisms of graphs and automata"  ''Univ. Michigan Technical Report'' , '''03105–44–T'''  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Hajnal,   "The chromatic number of the product of two $KAPAJA_1$-chromatic graphs can be countable"  ''Combinatorica'' , '''5'''  (1985)  pp. 137–140</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Lovász,   "Operations with structures"  ''Acta Math. Acad. Sci. Hung.''  (1967)  pp. 321–328</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.H. El-Zahar,   N.W. Sauer,   "The chromatic number of the product of two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110110/h11011072.png" />-chromatic graphs is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110110/h11011073.png" />"  ''Combinatorica'' , '''5''' :  2  (1985)  pp. 121–126</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Poljak,   "Coloring digraphs by iterated antichains"  ''Comment. Math. Univ. Carolin.'' , '''32''' :  2  (1991)  pp. 209–212</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Poljak,   V. Rödl,   "On the arc-chromatic number of a digraph"  ''JCT B'' , '''31'''  (1981)  pp. 190–198</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D.J. Miller,   "The categorical product of graphs"  ''Canad. J. Math.'' , '''20'''  (1968)  pp. 1511–1521</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Haggkvist,   P. Hell,   D.J. Miller,   V.N. Lara,   "On multiplicative graphs and the product conjecture"  ''Combinatorica'' , '''8''' :  1  (1988)  pp. 63–74</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D. Duffus,   N.W. Sauer,   "Lattices arising in categorial investigations of Hedetniemi's conjecture"  ''Discrete Math.'' , '''153'''  (1996)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  P. Hell,   H. Zhou,   X. Zhu,   "Homomorphisms to oriented cycles"  ''Combinatorica''  (??)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  N.W. Sauer,   X. Zhu,   "Multiplicative posets"  ''Order'' , '''8'''  (1992)  pp. 349–358</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  D. Duffus,   B. Sands,   R. Woodrow,   "On the chromatic number of the product of graphs"  ''J. Graph Th.'' , '''9'''  (1985)  pp. 487–495</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  N.W. Sauer,   X. Zhu,   "An approach to Hedetniemi's conjecture"  ''J. Graph Th.'' , '''16''' :  5  (1992)  pp. 423–436</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  G. Sabidussi,   "Graph multiplication"  ''Math. Z.'' , '''72'''  (1960)  pp. 446–457</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  R.J. Nowakowski,   D. Rall,   "Associative graph products and their independence, domination and coloring numbers"  ''J. Graph Th.''  (??)</TD></TR></table>