A note on Gallai–Ramsey number of even wheels

Abstract

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai k-coloring is a Gallai coloring that uses k colors. Given a graph H and an integer k≥1, the Gallai–Ramsey number GRk(H) of H is the least positive integer N such that every Gallai k-coloring of the complete graph KN contains a monochromatic copy of H. Let Wn denote a wheel on n+1 vertices. In this note, we study Gallai–Ramsey number of W2n and completely determine the exact value of GRk(W4) for all k≥2.