Upper bounds on the non-3-colourability threshold of random graphs

We present a full analysis of the expected number of 'rigid' 3-colourings of a sparse random graph. This shows that, if the average degree is at least 4.99, then as n → ∞ the expected number of such colourings tends to 0 and so the probability that the graph is 3-colourable tends to 0. (This result is tight, in that with average degree 4.989 the expected number tends to ∞.) This bound appears independently in Kaporis \textitet al. [Kap]. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989.

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

[2]  B. D. Bunday,et al.  Basic optimisation methods , 1985, Mathematical Gazette.

[3]  L. Kirousis,et al.  Approximating the unsatisfiability threshold of random formulas , 1998 .

[4]  M. J. Box.  A New Method of Constrained Optimization and a Comparison With Other Methods , 1965, Comput. J..

[5]  Michele Zito,et al.  An Inproved Upper Bound on the Non-3-Colourability Threshold , 1998, Inf. Process. Lett..

[6]  Colin McDiarmid,et al.  Surveys in Combinatorics, 1989: On the method of bounded differences , 1989 .

[7]  Yannis C. Stamatiou,et al.  A Note on the Non-Colorability Threshold of a Random Graph , 2000, Electron. J. Comb..

[8]  Yacine Boufkhad,et al.  A General Upper Bound for the Satisfiability Threshold of Random r-SAT Formulae , 1997, J. Algorithms.

[9]  Nico M. Temme,et al.  Asymptotic estimates of Stirling numbers , 1993 .

[10]  Michael Molloy,et al.  Almost all graphs with 2.522 n edges are not 3-colorable , 1999, Electron. J. Comb..

[11]  J. W. P. Hirschfeld,et al.  Thresholds for colourability and satisfiability in random graphs and boolean formulae , 2001 .

[12]  B. Bollobás.  The evolution of random graphs , 1984 .

[13]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[14]  Ehud Friedgut,et al.  A Sharp Threshold for k-Colorability , 1999, Random Struct. Algorithms.

[15]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[16]  Tad Hogg,et al.  The Hardest Constraint Problems: A Double Phase Transition , 1994, Artif. Intell..

[17]  A. Rbnyi.  ON THE EVOLUTION OF RANDOM GRAPHS , 2001 .