  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 $k$
colors. Given an integer $k\ge1$ and graphs $H_1, \ldots, H_k$, the
Gallai-Ramsey number $GR(H_1, \ldots, H_k)$ 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 $i \in \{1, \ldots, k\}$. When $H = H_1 =
\cdots = H_k$, we simply write $GR_k(H)$. We continue to study Gallai-Ramsey
numbers of even cycles and paths. For all $n\ge3$ and $k\ge1$, let
$G_i=P_{2i+3}$ be a path on $2i+3$ vertices for all $i\in\{0,1, \ldots, n-2\}$
and $G_{n-1}\in\{C_{2n}, P_{2n+1}\}$. Let $ i_j\in\{0,1,\ldots, n-1\}$ for all
$j\in\{1, \ldots, k\}$ with $ i_1\ge i_2\ge\cdots\ge i_k $. Song recently
conjectured that $GR(G_{i_1}, \ldots, G_{i_k}) = 3+\min\{i_1,
n^*-2\}+\sum_{j=1}^k i_j$, where $n^* =n$ when $G_{i_1}\ne P_{2n+1}$ and $n^*
=n+1$ when $G_{i_1}= P_{2n+1}$. This conjecture has been verified to be true
for $n\in\{3,4\}$ and all $k\ge1$. In this paper, we prove that the
aforementioned conjecture holds for $n \in\{5, 6\}$ and all $k \ge1$. Our
result implies that for all $k \ge 1$, $GR_k(C_{2n}) = GR_k(P_{2n}) =
(n-1)k+n+1$ for $n\in\{5,6\}$ and $GR_k(P_{2n+1})= (n-1)k+n+2$ for $1\le n \le6
$.
