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The expansion of half-integral polytopes

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arxiv 2402.14343 v1 pith:Y5M4RBAY submitted 2024-02-22 math.CO cs.CGcs.DMmath.MG

The expansion of half-integral polytopes

classification math.CO cs.CGcs.DMmath.MG
keywords expansionhalf-integralpolytopeszonotopesalwaysanalysisawaybounded
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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The expansion of a polytope is an important parameter for the analysis of the random walks on its graph. A conjecture of Mihai and Vazirani states that all $0/1$-polytopes have expansion at least 1. We show that the generalization to half-integral polytopes does not hold by constructing $d$-dimensional half-integral polytopes whose expansion decreases exponentially fast with $d$. We also prove that the expansion of half-integral zonotopes is uniformly bounded away from $0$. As an intermediate result, we show that half-integral zonotopes are always graphical.

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