Friendly bisections of random graphs

Resolving a conjecture of Füredi from 1988, we prove that with high probability, the random graph G(n, 1/2) admits a friendly bisection of its vertex set, i.e., a partition of its vertex set into two parts whose sizes differ by at most one in which n− o(n) vertices have at least as many neighbours in their own part as across. Our proof is constructive, and in the process, we develop a new method to study stochastic processes driven by degree information in random graphs; this involves combining enumeration techniques with an abstract second moment argument.

[1]  Asaf Ferber,et al.  Singularity of sparse random matrices: simple proofs , 2020, Comb. Probab. Comput..

[2]  Ronald D. Dutton,et al.  On satisfactory partitioning of graphs , 2002 .

[3]  Conrado Martínez,et al.  The MAX-CUT of sparse random graphs , 2012, SODA.

[4]  Nathan Linial,et al.  Asymptotically Almost Every 2r-Regular Graph Has an Internal Partition , 2017, Graphs and Combinatorics.

[5]  R. Sarpong,et al.  Bio-inspired synthesis of xishacorenes A, B, and C, and a new congener from fuscol† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sc02572c , 2019, Chemical science.

[6]  Luc Devroye,et al.  Local optima of the Sherrington-Kirkpatrick Hamiltonian , 2017, ArXiv.

[7]  Ashwin Sah,et al.  Majority Dynamics: The Power of One , 2021, 2105.13301.

[8]  Brendan D. McKay,et al.  Asymptotic enumeration of dense 0-1 matrices with specified line sums , 2008, J. Comb. Theory, Ser. A.

[9]  Tom Bohman,et al.  Random triangle removal , 2012, 1203.4223.

[10]  C. Newman,et al.  Local Minima in Disordered Mean-Field Ferromagnets , 2019, 1910.11862.

[11]  M. Talagrand.  The parisi formula , 2006 .

[12]  Alexander I. Barvinok,et al.  The number of graphs and a random graph with a given degree sequence , 2010, Random Struct. Algorithms.

[13]  Mihalis Yannakakis,et al.  Smoothed complexity of local max-cut and binary max-CSP , 2020, STOC.

[14]  O. Papaspiliopoulos.  High-Dimensional Probability: An Introduction with Applications in Data Science , 2020 .

[15]  Terence Tao,et al.  Additive combinatorics , 2007, Cambridge studies in advanced mathematics.

[16]  Anita Liebenau,et al.  Asymptotic enumeration of digraphs and bipartite graphs by degree sequence , 2020, Random Struct. Algorithms.

[17]  Linh Tran,et al.  Reaching a Consensus on Random Networks: The Power of Few , 2019, APPROX-RANDOM.

[18]  E. Bolthausen.  An estimate of the remainder in a combinatorial central limit theorem , 1984 .

[19]  Gonzalo Fiz Pontiveros,et al.  The triangle-free process and R(3,k) , 2013 .

[20]  Anita Liebenau,et al.  Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph , 2017, Journal of the European Mathematical Society.

[21]  Robert Morris,et al.  The Triangle-Free Process and the Ramsey Number 𝑅(3,𝑘) , 2020 .

[22]  Vojtech Rödl,et al.  On a Packing and Covering Problem , 1985, Eur. J. Comb..

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

[24]  Quan Li,et al.  On the max‐cut of sparse random graphs , 2014, Random Struct. Algorithms.

[25]  Nathan Linial,et al.  Internal Partitions of Regular Graphs , 2013, J. Graph Theory.

[26]  Peter Keevash,et al.  The early evolution of the H-free process , 2009, 0908.0429.

[27]  A. A. Serov,et al.  A Complete Proof of Universal Inequalities for the Distribution Function of the Binomial Law , 2013 .

[28]  N. Wormald.  ASYMPTOTIC ENUMERATION OF GRAPHS WITH GIVEN DEGREE SEQUENCE , 2019, Proceedings of the International Congress of Mathematicians (ICM 2018).

[29]  Brendan D. McKay,et al.  Asymptotic Enumeration by Degree Sequence of Graphs of High Degree , 1990, Eur. J. Comb..

[30]  C. Newman,et al.  Zero-Temperature Dynamics in the Dilute Curie–Weiss Model , 2017, Journal of Statistical Physics.

[31]  Carsten Thomassen.  Graph decomposition with constraints on the connectivity and minimum degree , 1983, J. Graph Theory.

[32]  Michael Stiebitz.  Decomposing graphs under degree constraints , 1996, J. Graph Theory.

[33]  Benny Sudakov,et al.  Anticoncentration for subgraph statistics , 2018, J. Lond. Math. Soc..

[34]  Michael U. Gerber,et al.  Algorithmic approach to the satisfactory graph partitioning problem , 2000, Eur. J. Oper. Res..

[35]  W. Marsden.  I and J , 2012 .

[36]  Robin Wilson,et al.  Modern Graph Theory , 2013 .

[37]  Yuval Peres,et al.  Local max-cut in smoothed polynomial time , 2017, STOC.

[38]  Béla Bollobás,et al.  Problems and results on judicious partitions , 2002, Random Struct. Algorithms.

[39]  E. C. Milner,et al.  Unfriendly partitions of a graph , 1990, J. Comb. Theory, Ser. B.