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To investigate the role of long-range van der Waals interactions in static friction, we derive an analytic expression for the coefficient of static friction $\mu_s$ between two thin layers of polarizable materials under zero load. For simplicity, we model the surface roughness with sinusoidal corrugations and calculate the interaction energy perturbatively up to the second order in corrugation amplitude. The ratio of corresponding maximum lateral Casimir force to normal Casimir force is defined as the coefficient of static friction, which is found to be independent of the dielectric properties of the materials. It depends on the geometric properties, like interlayer separation, corrugation amplitude, and wavelength of the corrugation. As a proof of concept, our predicted values of $\mu_s$ for the 2D van der Waals materials graphene and hexagonal boron nitride are in reasonable agreement with previously reported values in the literature. This simplistic model could be generalized by incorporating other forces, such as the frequency-dependent contributions of van der Waals interactions and electrostatic interactions., Comment: 11 Pages, 7 Figures, 1 Table

We show that the problem of deciding whether the vertex set of a graph can be covered with at most two bicliques is in NP$\cap$coNP. We thus almost determine the computational complexity of a problem whose status has remained open for quite some time. Our result implies that a polynomial time algorithm for the problem is more likely than it being NP-complete unless P = NP.