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2013 IEEE 54th Annual Symposium on Foundations of Computer Science (2013)
Berkeley, CA USA
Oct. 26, 2013 to Oct. 29, 2013
ISSN: 0272-5428
pp: 380-389
Amin Coja-Oghlan , Math. Inst., Goethe Univ., Frankfurt, Germany
Dan Vilenchik , Facutly of Math. & Comput. Sci, Weizamnn Inst., Rehovot, Israel
In this paper we establish a substantially improved lower bound on the k-color ability threshold of the random graph G(n, m) with n vertices and m edges. The new lower bound is ≈ 1.39 less than the 2k ln (k)-ln (k) first-moment upper bound (and approximately 0.39 less than the 2k ln (k) - ln(k) - 1 physics conjecture). By comparison, the best previous bounds left a gap of about 2+ln(k), unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004]. Furthermore, we prove that, in a precise sense, our lower bound marks the so-called condensation phase transition predicted on the basis of physics arguments [Krzkala et al.: PNAS 2007]. Our proof technique is a novel approach to the second moment method, inspired by physics conjectures on the geometry of the set of k-colorings of the random graph.
Physics, Method of moments, Color, Geometry, Cavity resonators, Random variables, Optimization

A. Coja-Oghlan and D. Vilenchik, "Chasing the K-Colorability Threshold," 2013 IEEE 54th Annual Symposium on Foundations of Computer Science(FOCS), Berkeley, CA USA, 2014, pp. 380-389.
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