A note on Hedetniemi's conjecture, Stahl's conjecture and the Poljak-R\"odl function

We prove that $\min\{\chi(G), \chi(H)\} - \chi(G\times H)$ can be arbitrarily large, and that if Stahl's conjecture on the multichromatic number of Kneser graphs holds, then $\min\{\chi(G), \chi(H)\}/\chi(G\times H) \leq 1/2 + \epsilon$ for large values of $\min\{\chi(G), \chi(H)\}$.

The Poljak-Rödl function f : N → N is defined by Hedetniemi's conjecture is equivalent to the statement that f (n) = n for all n. Shitov proved that for sufficiently large n, f (n) ≤ n − 1. Still, very little is known about the behavior of the function f (n). In particular, it is unknown whether f (n) is bounded by a constant. However it is known that if f (n) is bounded by a constant, then f (n) ≤ 9 for all n (see [10,14]). In this note, we prove the following facts.
Proposition 1 (i) will be proved in Section 2. Proposition 1 (ii) will be proved in Section 3, where a presentation of Stahl's conjecture is also given.

Discussion and extensions of Shitov's results
For a positive integer c, the exponential graph K H c has vertices all the mappings for every edge e = uv of H. It is well known and easy to verify that Φ Thus the way to find counterexamples to Hedetniemi's conjecture is to find an integer c and a graph H such that both H and K H c have chromatic number larger than c.
Shitov's construction of counterexamples to Hedetniemi's conjecture is based on the following result.

Theorem 2 ([12], Claim 3).
For any graph G with girth at least six, for all but finitely many values of q, we have χ K Finding such a lower bound on chromatic numbers of some exponential graphs was the key part of Shitov's refutation of Hedetniemi's conjecture. Finding lexicographic products G[K q ] with χ(G[K q ]) > c is standard theory. Indeed the fractional chromatic number χ f (H) of a graph H is a standard lower bound for its chromatic number, and it is well known that [3]). Erdős' classic probabilistic proof [2] shows that there are graphs with girth at least 6 and fractional chromatic number at least 3.1. For such a graph G, we have , and by Theorem 2, this yields a counterexample to Hedetniemi's conjecture.
Remarkably, replacing the condition χ f (G) ≥ 3.1 by χ f (G) ≥ B for B ≫ 3.1 readily gives counterexamples to Hedetniemi's conjecture where the chromatic number of at least one factor is arbitrarily larger than the chromatic number of the product. Also, the proof of Theorem 2 only uses a small subgraph of K Therefore it is possible that Shitov's construction already gives examples that show that lim n→∞ f (n)/n = 0. On the other hand, since χ f (G[K q ]) > c, the fractional version of Hedetniemi's conjecture [15] implies that χ f (K G[Kq] c ) = c. Thus it is also reasonnable to think that χ(K

G[Kq] c
)/c may be bounded, and that the identity lim n→∞ f (n)/n = 0, if true, can only be witnessed by a different construction.
Proof of Proposition 1 (i). Fix a positive integer d. We shall prove that if n is sufficiently large, then f (n + d) ≤ n. Let G d be a graph with girth at least 6 and fractional chromatic number at least 8d. Then by Theorem 2, for sufficiently large q and c = ⌈3.1q⌉, we have χ K and hence at least c + 1 colours are needed for each copy. For i = j, each function in Q i is adjacent to each function in Q j . Hence, Thus for every d there exist infinitely many values of n (of the form dc + d) such that n − f (n) ≥ d. It only remains to show that the gap between n and f (n) will not close while going from one value of c to the next. Note that c = ⌈3.1q⌉, where q is any value above a fixed threshold, and ⌈3.1(q + 1)⌉ − ⌈3.1q⌉ ≤ 4. Thus it suffices to examine the values n = dc + d + i where i ≤ 4d, and we can suppose that c ≥ 5. The graph K Altogether, the inequality f (n + d) ≤ n is established for all but finitely many values of n. Thus, lim n→∞ n − f (n) = ∞.
The gap between n and f (n) proved in this section depends on the minimum number p of vertices of a girth 6 graph with fractional chromatic number at least 8d. The best known upper bound for p to our knowledge is p = O((d log d) 4 ), which follows from a result of Krivelevich [8]. Using this result, one can show that for any ǫ > 0, there is a constant a such that for sufficiently large n, f (n) ≤ n − a(log n) 1/4−ǫ . Very recently, He and Wigderson [5] proved that for some ǫ ≃ 10 −9 , f (n) < (1 − ǫ)n for sufficiently large n. The examples are again cases of Shitov's construction.

Stahl's conjecture
In the proof of Proposition 1(i), based on the fact that χ(K . This property can be formulated in terms of homomorphisms of Kneser graphs. Recall that the vertices of the Kneser graph K(m, n) are the n-subsets of {1, . . . , m}, and two of these are joined by an edge whenever they are disjoint. Thus the colouring φ : K → K x induces a homomorphism ψ : K(cd, c) → K(x, c + 1). The question is how large does x need to be for such a homomorphism to exist.
Stahl's conjecture deals with the latter question. For an integer n, the n-th multichromatic number χ n (H) of a graph H is the least integer m such that H admits a homomorphism to K(m, n). In particular χ 1 (H) = χ(H). Lovász [9] proved that χ 1 (K(m, n)) = χ(K(m, n)) = m − 2n + 2. Stahl [11] investigated the general multichromatic numbers of Kneser graphs, and observed the following.
Proof of Proposition 1 (ii). For a fixed d, let G d have girth at least 6 and fractional chromatic number at least 8d. For any q above a given threshold q d and for c = ⌈3.1q⌉, we have χ(G d [K q ]) ≥ 2cd and χ K Since this holds for arbitrarily large d, lim sup f (n) n ≤ 1 2 .