Better bounds for poset dimension and boxicity

04/09/2018

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by   Alex Scott, et al.

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We prove that the dimension of every poset whose comparability graph has maximum degree Δ is at most Δ^1+o(1)Δ. This result improves on a 30-year old bound of Füredi and Kahn, and is within a ^o(1)Δ factor of optimal. We prove this result via the notion of boxicity. The boxicity of a graph G is the minimum integer d such that G is the intersection graph of d-dimensional axis-aligned boxes. We prove that every graph with maximum degree Δ has boxicity at most Δ^1+o(1)Δ, which is also within a ^o(1)Δ factor of optimal. We also show that the maximum boxicity of graphs with Euler genus g is Θ(√(g g)), which solves an open problem of Esperet and Joret and is tight up to a O(1) factor.