A conjecture on Gallai-Ramsey numbers of even cycles and paths

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 Gallai- Ramsey 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 H-i in color i for some i is an element of {1, 2, ..., k}. When H = H-1 = ... = H-k, we simply write GR(k) (H). We study Gallai-Ramsey numbers of even cycles and paths. For all n >= 3 and k >= 2, 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) is an element of {C-2n, P2n+1} Let i(j) is an element of {0, 1, ..., n - 1} for all j is an element of {1, 2, ..., k} with i(1) >= i(2) >= ... >= i(k). The first author recently conjectured that GR(G(i1), G(i2), ..., G(ik)) = vertical bar G(i1)vertical bar + Sigma(k)(j=2) i(j). The truth of this conjecture implies that GR(k)(C-2n) = GR(k)(P-2n) = (n - 1)k +n+1 for all n >= 3 and k >= 1, and GR(k) (P2n+1) = (n - 1)k + n + 2 for all n >= 1 and k >= 1. In this paper, we prove that the aforementioned conjecture holds for n is an element of {3, 4} and all k >= 2. Our proof relies only on Gallai's result and the classical Ramsey numbers R(H-1, H-2), where H-1, H-2 is an element of {C-8, C-6, P-7, P-5, P-3} . We believe the recoloring method we develop here will be very useful for solving subsequent cases, and perhaps the conjecture.