The Chromatic Number of Random Graphs for Most Average Degrees

For a fixed number d> 0 and n large, let G(n,d/n) be the random graph on n vertices in which any two vertices are connected with probability d/n independently. The problem of determining the chromatic number of G(n,d/n) goes back to the famous 1960 article of Erdős and Renyi that started the theory of random graphs [Magayar Tud. Akad. Mat. Kutato Int. Kozl. 5 (1960) 17–61]. Progress culminated in the landmark paper of Achlioptas and Naor [Ann. Math. 162 (2005) 1333–1349], in which they calculate the chromatic number precisely for all d in a set S⊂ (0,∞) of asymptotic density limz→∞ z ∫z 0 1S = 2 , and up to an additive error of one for the remaining d. Here we obtain a near-complete answer by determining the chromatic number of G(n,d/n) for all d in a set of asymptotic density 1.

[1]  Martin E. Dyer,et al.  On the chromatic number of a random hypergraph , 2015, J. Comb. Theory, Ser. B.

[2]  Cristopher Moore,et al.  Random k-SAT: Two Moments Suffice to Cross a Sharp Threshold , 2003, SIAM J. Comput..

[3]  Cristopher Moore,et al.  The Chromatic Number of Random Regular Graphs , 2004, APPROX-RANDOM.

[4]  Florent Krzakala,et al.  Phase Transitions in the Coloring of Random Graphs , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Florent Krzakala,et al.  Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Benny Sudakov,et al.  Coloring Random Graphs , 1998, Inf. Process. Lett..

[7]  K. Appel,et al.  Every planar map is four colorable. Part I: Discharging , 1977 .

[8]  Cristopher Moore,et al.  Tight Bounds on the Threshold for Permuted k-Colorability , 2011, APPROX-RANDOM.

[9]  Amin Coja-Oghlan,et al.  Upper-Bounding the k-Colorability Threshold by Counting Covers , 2013, Electron. J. Comb..

[10]  Béla Bollobás,et al.  The chromatic number of random graphs , 1988, Comb..

[11]  Robin Thomas,et al.  The Four-Colour Theorem , 1997, J. Comb. Theory, Ser. B.

[12]  Béla Bollobás,et al.  Random Graphs , 1985 .

[13]  Joel H. Spencer,et al.  Sharp concentration of the chromatic number on random graphsGn, p , 1987, Comb..

[14]  Michael Molloy,et al.  The analysis of a list-coloring algorithm on a random graph , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[15]  Michael Krivelevich,et al.  Coloring Random Graphs — an Algorithmic Perspective , 2002 .

[16]  Konstantinos Panagiotou,et al.  Catching the k-NAESAT threshold , 2011, STOC '12.

[17]  Noga Alon,et al.  The concentration of the chromatic number of random graphs , 1997, Comb..

[18]  D. Achlioptas,et al.  A sharp threshold for k-colorability , 1999 .

[19]  T. Toffoli.  Physics and computation , 1982 .

[20]  Lenka Zdeborová,et al.  The condensation transition in random hypergraph 2-coloring , 2011, SODA.

[21]  G. Grimmett,et al.  On colouring random graphs , 1975 .

[22]  Amin Coja-Oghlan,et al.  On the chromatic number of random regular graphs , 2016, J. Comb. Theory, Ser. B.

[23]  M. Mézard,et al.  Information, Physics, and Computation , 2009 .

[24]  K. Appel,et al.  Every Planar Map Is Four Colorable , 2019, Mathematical Solitaires & Games.

[25]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[26]  D. Saad,et al.  Random graph coloring: statistical physics approach. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[28]  Colin McDiarmid,et al.  Algorithmic theory of random graphs , 1997 .

[29]  Tomasz Luczak.  A note on the sharp concentration of the chromatic number of random graphs , 1991, Comb..

[30]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[31]  Tomasz Luczak.  The chromatic number of random graphs , 1991, Comb..

[32]  D. W. MATULA.  Expose-and-merge exploration and the chromatic number of a random graph , 1987, Comb..

[33]  Amin Coja-Oghlan,et al.  Algorithmic Barriers from Phase Transitions , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[34]  Béla Bollobás,et al.  The scaling window of the 2‐SAT transition , 1999, Random Struct. Algorithms.

[35]  A. Naor,et al.  The two possible values of the chromatic number of a random graph , 2005 .

[36]  Andrea Montanari,et al.  Gibbs states and the set of solutions of random constraint satisfaction problems , 2006, Proceedings of the National Academy of Sciences.

[37]  N. Wormald,et al.  On the chromatic number of random d-regular graphs , 2008, 0812.2937.

[38]  Alan M. Frieze,et al.  Random k-Sat: A Tight Threshold For Moderately Growing k , 2005, Comb..

[39]  V. Bapst,et al.  The Condensation Phase Transition in Random Graph Coloring , 2016 .

[40]  Riccardo Zecchina,et al.  Coloring random graphs , 2002, Physical review letters.

[41]  M. Mézard,et al.  Analytic and Algorithmic Solution of Random Satisfiability Problems , 2002, Science.

[42]  Noga Alon,et al.  A Spectral Technique for Coloring Random 3-Colorable Graphs , 1997, SIAM J. Comput..

[43]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[44]  Konstantinos Panagiotou,et al.  Going after the k-SAT threshold , 2013, STOC '13.