Gallai–Ramsey number of odd cycles with chords

A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai k-coloring is a Gallai coloring that uses at most k colors. For an integer k≥1, the Gallai–Ramsey number GRk(H) of a given 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. 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. We prove that GRk(Θ2n+1)=n⋅2k+1 for all k≥1 and n≥3. This implies that GRk(C2n+1)=n⋅2k+1 all k≥1 and n≥3. Our result yields a unified proof for the Gallai–Ramsey number of all odd cycles on at least five vertices.