Abstract
Hedetniemi conjectured in 1966 that Hedetniemi conjectured in 1966 that \(\chi(G \times H) = \min\{\chi(G), \chi(H)\}\) for any graphs G and H. Here \(G\times H\) is the graph with vertex set \(V(G)\times V(H)\) defined by putting \((x,y)\) and \((x^{\prime}, y^{\prime})\) adjacent if and only if \(xx^{\prime}\in E(G)\) and \(yy^{\prime}\in V(H)\). This conjecture received a lot of attention in the past half century. It was disproved recently by Shitov. The Poljak-Rodl function is defined as \(f(n) = \min\{\chi(G \times H): \chi(G)=\chi(H)=n\}\). Hedetniemi's conjecture is equivalent to saying \(f(n)=n\) for every integer \(n\). Shitov’s result shows that \(f(n)<n\) when \(n\) is sufficiently large. Using Shitov’s result, Tardif and Zhu showed that \(f(n) \le n - (\log n)^{1/4-o(1)}\) for sufficiently large \(n\). Using Shitov’s method, He and Wigderson showed that for \(\epsilon \approx 10^{-9}\) and \(n\) sufficiently large, \(f(n) \le (1-\epsilon)n\). In this note we observe that a slight modification of the proof in the paper of Zhu and Tardif shows that \(f(n) \le (\frac 12 + o(1))n\) for sufficiently large \(n\). On the other hand, it is unknown whether \(f(n)\) is bounded by a constant. However, we do know that if \(f(n)\) is bounded by a constant, then the smallest such constant is at most 9. This note gives self-contained proofs of the above mentioned results.
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Zhu, X. (2021). Note on Hedetniemi’s Conjecture and the Poljak-Rödl Function. In: de Gier, J., Praeger, C.E., Tao, T. (eds) 2019-20 MATRIX Annals. MATRIX Book Series, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-030-62497-2_31
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