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  "body": "\n## Counterexample\n\nLet $G$ and $H$ be finite [[graphs|0JD5C82QMB]].\nTheir tensor product $G \\times H$ has vertex set $V(G) \\times V(H)$.\nIn it, $(g, h)$ is adjacent to $(g', h')$ when $gg'$ is an edge of $G$ and $hh'$ is an edge of $H$.\nHedetniemi's conjecture asserted that $\\chi(G \\times H) = \\min(\\chi(G), \\chi(H))$ for all finite graphs $G$ and $H$, where $\\chi$ is the [[chromatic number|0C5M7AM6FH]].\nThe conjecture is false: in 2019 Yaroslav Shitov constructed finite graphs $G$ and $H$ with\n$$\\chi(G \\times H) \u003c \\min(\\chi(G), \\chi(H)).$$\n\n## Remarks\n\nThe inequality $\\chi(G \\times H) \\le \\min(\\chi(G), \\chi(H))$ always holds.\nIndeed, the projection $(g, h) \\mapsto g$ is a [[graph homomorphism|06SCB3R6BN]] to $G$, so a proper colouring of $G$ pulls back to one of $G \\times H$.\nThe same holds for $H$.\nShitov's counterexample uses an exponential graph.\nFor a graph $G$ and $c \\ge 1$, the vertices of $K_c^G$ are all maps $f \\colon V(G) \\to \\set{1, \\dots, c}$.\nTwo maps $f$ and $f'$ are adjacent when $f(u) \\ne f'(v)$ for every edge $uv$ of $G$, in both orders of $u$ and $v$.\nThe map $(v, f) \\mapsto f(v)$ is a proper colouring of $G \\times K_c^G$ with $c$ colours.\nIndeed, if $(u, f)$ and $(v, f')$ are adjacent, then $uv$ is an edge and $f$ is adjacent to $f'$, so $f(u) \\ne f'(v)$.\nSo any $G$ with $\\chi(G) \u003e c$ and $\\chi(K_c^G) \u003e c$ is a counterexample, and Shitov showed that such $G$ exist for large $c$.\nThe conjecture holds when $\\min(\\chi(G), \\chi(H)) \\le 4$; the case $4$ is a theorem of El-Zahar and Sauer.\n\n## Explanations\n\n### L1 human\n\nIn the tensor product of two graphs, two pairs are joined when both coordinates are joined.\nColouring either factor gives a colouring of the product, so the product never needs more colours than the easier factor.\nFor over fifty years people believed it never needs fewer, but Shitov found very large graphs where it does.\n",
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