Gallai-Ramsey number of even cycles with chords

Abstract

For a graph H and an integer k≥1, the k-color Ramsey number Rk(H) is the least integer N such that every k-coloring of the edges of the complete graph KN contains a monochromatic copy of H. Let Cm denote the cycle on m≥4 vertices and let Θm denote the family of graphs obtained from Cm by adding an additional edge joining two non-consecutive vertices. Unlike Ramsey number of odd cycles, little is known about the general behavior of Rk(C2n) except that Rk(C2n)≥(n−1)k+n+k−1 for all k≥2 and n≥2. In this paper, we study Ramsey number of even cycles with chords under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. For an integer k≥1, the Gallai-Ramsey number GRk(H) of a graph H is the least positive integer N such that every Gallai k-coloring of the complete graph KN contains a monochromatic copy of H. We prove that GRk(Θ2n)=(n−1)k+n+1 for all k≥2 and n≥3. This implies that GRk(C2n)=(n−1)k+n+1 all k≥2 and n≥3. Our result yields a unified proof for the Gallai-Ramsey number of all even cycles on at least four vertices.