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Keywords:

  • rainbow triangle;
  • Gallai coloring;
  • path;
  • cycle;
  • Gallai Ramsey

Abstract

Given a graph G and a positive integer k, define the Gallai–Ramsey number to be the minimum number of vertices n such that any k-edge coloring of inline image contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we improve upon known upper bounds on the Gallai–Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of magnitude as functions of k.