论文标题
连接的大小拉姆齐数量的比赛与小路径或周期
Connected size Ramsey numbers of matchings versus a small path or cycle
论文作者
论文摘要
给定两个图$ g_1，g_2 $，连接的大小ramsey编号$ {\ hat {r}} _ c（g_1，g_2）$被定义为连接的图形$ g $的最小边数，因此，对于任何$ g_ $ g_ $ g_ $ g_ $ g_ $ g_2 $ g_的红色蓝色边缘颜色来说，是$ g_ $ g_ $ g_ $ g_ $ g_2 $ $ g_2 $。集中精力$ {\ hat {r}} _ c（nk_2，g_2）$，其中$ nk_2 $是一种匹配，我们概括并改善两个先前的结果，如下所示。 Vito，Nabila，Safitri和Silaban获得了$ {\ hat {r}} _ C（nk_2，p_3）$的确切值，$ n = 2,3,4 $。我们确定其所有正整数$ n $的确切值。 Rahadjeng，Baskoro和Assiyatun证明了$ {\ hat {r}} _ c（nk_2，c_4）\ le 5n-1 $ for $ n \ ge 4 $。我们将上限从$ 5N-1 $提高到$ \ lfloor（9n-1）/2 \ rfloor $。此外，我们显示的结果具有相同的味道，并且具有精确的值：$ {\ hat {r}} _ c（nk_2，c_3）= 4n-1 $ for laster Integers $ n $。
Given two graphs $G_1, G_2$, the connected size Ramsey number ${\hat{r}}_c(G_1,G_2)$ is defined to be the minimum number of edges of a connected graph $G$, such that for any red-blue edge colouring of $G$, there is either a red copy of $G_1$ or a blue copy of $G_2$. Concentrating on ${\hat{r}}_c(nK_2,G_2)$ where $nK_2$ is a matching, we generalise and improve two previous results as follows. Vito, Nabila, Safitri, and Silaban obtained the exact values of ${\hat{r}}_c(nK_2,P_3)$ for $n=2,3,4$. We determine its exact values for all positive integers $n$. Rahadjeng, Baskoro, and Assiyatun proved that ${\hat{r}}_c(nK_2,C_4)\le 5n-1$ for $n\ge 4$. We improve the upper bound from $5n-1$ to $\lfloor (9n-1)/2 \rfloor$. In addition, we show a result which has the same flavour and has exact values: ${\hat{r}}_c(nK_2,C_3)=4n-1$ for all positive integers $n$.
