Gallai–Ramsey number of odd cycles with chords

A Gallai coloring is a of the edges complete graph without rainbow triangles, and k-coloring that uses at most k colors. For an integer k≥1, Gallai–Ramsey number GRk(H) given H least positive N such every KN contains monochromatic copy H. Let Cm denote cycle on m≥4 vertices let Θm family graphs obtained from by adding additional edge joining two non-consecutive vertices. We prove GRk(Θ2n+1)=n⋅2k+1 for all k≥1 n≥3. This implies GRk(C2n+1)=n⋅2k+1 Our result yields unified proof odd cycles five