论文标题
具有五个顶点和八个边缘的图形的Gallai-Ramsey编号
Gallai-Ramsey numbers for graphs with five vertices and eight edges
论文作者
论文摘要
Gallai $ k $ - 颜色是$ k $ - 边缘的完整图形，其中没有彩虹三角形。对于给定图，$ g_1，g_2，g_3 $和非负整数$ r，s，t $带有$ k = r+s+s+t $，$ k $ -colored colored gallai-ramsey number $ gr_ gr_ {k_ {k_ {k_ {3}每个Gallai $ k $颜色$ k_ {n} $包含$ g_1 $的单色副本，由第一个$ r $颜色之一或单色copy $ g_2 $的单色副本，由中间$ s $颜色之一或单色颜色之一或单色copy of $ g_3 $，由$ g_3 $ coled $ g_3 $，由$ g_3 $颜色为$ t $ t $ t $ t $ t $ t $。在本文中，我们确定了加莱 - 拉姆西号的值，如果$ g_1 = b_1 = b_ {3}^{+} $，$ g_2 = s_2 = s_ {3}^+$和$ g_3 = k_3 $。然后获得Gallai-Ramsey编号$ gr_ {k {3}：b_ {3}^{+}）$。因此，完全求解了具有五个顶点和八个边缘的图形的Gllai-Ramsey编号。此外，Gallai-ramsey编号$ gr_ {k}（k_ {3}：r \ cdot b_3^+，〜（k-r）\ cdot s_3^+）$，$ gr_ {k_ {k_ {3}： $ gr_ {k_ {3}：s \ cdot s_3^+，〜（k-s）\ cdot k_3）$尊重。
A Gallai $k$-coloring is a $k$-edge coloring of a complete graph in which there are no rainbow triangles. For given graphs $G_1, G_2, G_3$ and nonnegative integers $r, s, t$ with that $k=r+s+t$, the $k$-colored Gallai-Ramsey number $gr_{k}(K_{3}: r\cdot G_1,~ s\cdot G_2, ~t\cdot G_3)$ is the minimum integer $n$ such that every Gallai $k$-colored $K_{n}$ contains a monochromatic copy of $G_1$ colored by one of the first $r$ colors or a monochromatic copy of $G_2$ colored by one of the middle $s$ colors or a monochromatic copy of $G_3$ colored by one of the last $t$ colors. In this paper, we determine the value of Gallai-Ramsey number in the case that $G_1=B_{3}^{+}$, $G_2=S_{3}^+$ and $G_3=K_3$. Then the Gallai-Ramsey number $gr_{k}(K_{3}: B_{3}^{+})$ is obtained. Thus the Gllai-Ramsey numbers for graphs with five vertices and eight edges are solved completely. Furthermore, the the Gallai-Ramsey numbers $gr_{k}(K_{3}: r\cdot B_3^+,~ (k-r)\cdot S_3^+)$, $gr_{k}(K_{3}: r\cdot B_3^+,~ (k-r)\cdot K_3)$ and $gr_{k}(K_{3}: s\cdot S_3^+,~ (k-s)\cdot K_3)$ are obtained, respecticely.
