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.

翻译：图$H$被称为公共的，若在红色/蓝色边着色的大完全图中，单色标号副本$H$的数量渐近地被每种颜色等比例随机着色所最小化。我们将此概念扩展到非对称设定。即，定义图对$(H_1,H_2)$为$(p,1-p)$-公共的，若红色中$H_1$密度与蓝色中$H_2$密度的特定线性组合渐近地被每条边以概率$p$染红、以概率$1-p$染蓝的随机着色所最小化。我们将公共图的许多结果推广到这种非对称设定。此外，我们获得了若干关于图公共对的新结果，这些结果在对称设定中没有自然对应项。我们还得到了经典意义下公共图的新例子，并提出了几个开放问题。