On the Power of Choice for k-Colorability of Random Graphs

Authors Varsha Dani, Diksha Gupta, Thomas P. Hayes

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Author Details

Varsha Dani
  • Ronin Institute, Montclair, NJ, USA
  • Dept. of Computer Science, Rochester Institute of Technology, NY, USA
Diksha Gupta
  • School of Computing, National University of Singapore, Singapore
Thomas P. Hayes
  • Dept. of Computer Science, University of New Mexico, Albuquerque, NM, USA
  • http:cs.unm.edu/ hayest


The authors would like to thank Will Perkins for suggesting the problem of shifting the threshold for k-coloring, and Will Perkins and Cris Moore for helpful conversations.

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Varsha Dani, Diksha Gupta, and Thomas P. Hayes. On the Power of Choice for k-Colorability of Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 59:1-59:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In an r-choice Achlioptas process, random edges are generated r at a time, and an online strategy is used to select one of them for inclusion in a graph. We investigate the problem of whether such a selection strategy can shift the k-colorability transition; that is, the number of edges at which the graph goes from being k-colorable to non-k-colorable. We show that, for k ≥ 9, two choices suffice to delay the k-colorability threshold, and that for every k ≥ 2, six choices suffice.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Stochastic processes
  • Theory of computation → Generating random combinatorial structures
  • Random graphs
  • Achlioptas Processes
  • Phase Transition
  • Graph Colorability


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