Gallai-Ramsey numbers of C10 and C12

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. Given an integer k >= 1 and graphs H-1,..., H-k, the GallaiRamsey number GR(H-1,..., H-k) is the least integer n such that every Gallai k-coloring of the complete graph K-n contains a monochromatic copy of Hi in color i for some i is an element of{1,..., k}. When H = H-1 = . . . = H-k, we simply write GR(k)(H). We continue to study Gallai-Ramsey numbers of even cycles and paths. For all n >= 3 and k = 1, let G(i) = P2i+ 3 be a path on 2i + 3 vertices for all i is an element of {0, 1,..., n - 2} and G(n-1). {C-2n, P2n+ 1}. Let i(j) is an element of {0, 1,..., n - 1} for all j is an element of {1,..., k} with i(1) >= i(2) >= . . . >= i(k). Song recently conjectured that GR(Gi(1),..., Gi(k)) = vertical bar G(i1)vertical bar + Sigma(k)(j=2) i(j). This conjecture has been verified to be true for n is an element of {3, 4} and all k = 1. In this paper, we prove that the aforementioned conjecture holds for n is an element of{5, 6} and all k = 1. Our result implies that for all k >= 1, GR(k)(C-2n) = GR(k)(P-2n) = (n- 1)k+ n+ 1 for n is an element of {5, 6} and GR(k)(P2n+1) = (n- 1)k+ n + 2 for 1 <= n <= 6.