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A classic result on the gap between list chromatic number and the chromatic number tells us $\\chi_\\ell(K_{a,b}) = 1 + a$ if and only if $b \\geq a^a$. Since $\\chi_\\ell(K_{a,b}) \\leq 1 + a$ for any $b \\in \\mathbb{N}$, this result tells us the values of $b$ for which $\\chi_\\ell(K_{a,b})$ is as large as possible and far from $\\chi(K_{a,b})=2$. 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Mudrock","submitted_at":"2018-11-02T18:30:42Z","abstract_excerpt":"We study the list chromatic number of the Cartesian product of any graph $G$ and a complete bipartite graph with partite sets of size $a$ and $b$, denoted $\\chi_\\ell(G \\square K_{a,b})$. We have two motivations. A classic result on the gap between list chromatic number and the chromatic number tells us $\\chi_\\ell(K_{a,b}) = 1 + a$ if and only if $b \\geq a^a$. Since $\\chi_\\ell(K_{a,b}) \\leq 1 + a$ for any $b \\in \\mathbb{N}$, this result tells us the values of $b$ for which $\\chi_\\ell(K_{a,b})$ is as large as possible and far from $\\chi(K_{a,b})=2$. 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