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Currently, the best known upper bound for $\\chi_{\\ell}(G)$ is $(1 + o(1)) \\frac{\\Delta}{\\log \\Delta}$, which also holds for the much larger class of triangle-free graphs. We prove that for $\\varepsilon = 10^{-3}$, every bipartite graph $G$ of sufficiently large maximum degree $\\Delta$ satisfies $\\chi_{\\ell}(G) < (\\frac{4}{5} -\\varepsilon) \\frac{\\Delta}{\\log \\Delta}$. 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Currently, the best known upper bound for $\\chi_{\\ell}(G)$ is $(1 + o(1)) \\frac{\\Delta}{\\log \\Delta}$, which also holds for the much larger class of triangle-free graphs. We prove that for $\\varepsilon = 10^{-3}$, every bipartite graph $G$ of sufficiently large maximum degree $\\Delta$ satisfies $\\chi_{\\ell}(G) < (\\frac{4}{5} -\\varepsilon) \\frac{\\Delta}{\\log \\Delta}$. 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