Colouring locally sparse graphs with the first moment method
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We give a short proof of a bound on the list chromatic number of graphs G of maximum degree Δ where each neighbourhood has density at most d, namely χ_ℓ(G) ≤ (1+o(1)) Δ/lnΔ/d+1 as Δ/d+1→∞. This bound is tight up to an asymptotic factor 2, which is the best possible barring a breakthrough in Ramsey theory, and strengthens results due to Vu, and more recently Davies, P., Kang, and Sereni. Our proof relies on the first moment method, and adapts a clever counting argument developed by Rosenfeld in the context of non-repetitive colourings. As a final touch, we show that our method provides an asymptotically tight lower bound on the number of colourings of locally sparse graphs.
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