TY - GEN

T1 - Chasing the k-colorability threshold

AU - Coja-Oghlan, Amin

AU - Vilenchik, Dan

PY - 2013/12/1

Y1 - 2013/12/1

N2 - In this paper we establish a substantially improved lower bound on the k-colorability 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 ≈ 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 + lnk, 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.

AB - In this paper we establish a substantially improved lower bound on the k-colorability 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 ≈ 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 + lnk, 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.

KW - Graph coloring

KW - Phase transitions

KW - Random structures

UR - http://www.scopus.com/inward/record.url?scp=84893448081&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2013.48

DO - 10.1109/FOCS.2013.48

M3 - Conference contribution

AN - SCOPUS:84893448081

SN - 9780769551357

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 380

EP - 389

BT - Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013

T2 - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013

Y2 - 27 October 2013 through 29 October 2013

ER -