论文标题
图表的共同对
Common Pairs of Graphs
论文作者
论文摘要
如果大型完整图的红色/蓝色边缘着色的单色标记副本的数量渐近地最小化，每种颜色的随机着色渐近地最小化了图形$ h $，则据说是常见的。我们将此概念扩展到不对称的设置。也就是说，我们将图形$（H_1，H_2）$定义为$（p，1-p）$ - 如果特定的线性组合$ h_1 $ in Red和$ h_2 $ in Red and $ h_2 $的蓝色密度的特定线性组合是由随机颜色渐近地最小化的，每种边缘与可能性$ p $ p $ p $ p $ p $ p $ and Blue ablesability cobyability cobibality pobibality $ 1-P $ 1-P $ 1-P $ 1-P $ 1-P $ 1-P $。我们将许多结果扩展到通用图上的许多结果。此外，我们为在对称环境中没有天然模拟的常见图表获得了几个新的结果。我们还从经典意义上获得了常见图形的新示例，并提出了几个开放问题。
A graph $H$ is said to be common if the number of monochromatic labelled copies of $H$ in a red/blue edge colouring of a large complete graph is asymptotically minimized by a random colouring with an equal proportion of each colour. We extend this notion to an asymmetric setting. That is, we define a pair $(H_1,H_2)$ of graphs to be $(p,1-p)$-common if a particular linear combination of the density of $H_1$ in red and $H_2$ in blue is asymptotically minimized by a random colouring in which each edge is coloured red with probability $p$ and blue with probability $1-p$. We extend many of the results on common graphs to this asymmetric setting. In addition, we obtain several novel results for common pairs of graphs with no natural analogue in the symmetric setting. We also obtain new examples of common graphs in the classical sense and propose several open problems.
