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Common pairs of graphs
Part of:
Graph theory
Published online by Cambridge University Press: 30 June 2025
Abstract
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 minimised by a random colouring in which each edge is equally likely to be red or blue. We extend this notion to an off-diagonal 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 minimised by a random colouring in which each edge is coloured red with probability 
$p$ and blue with probability 
$1-p$. Our results include off-diagonal extensions of several standard theorems on common graphs and novel results for common pairs of graphs with no natural analogue in the classical setting.
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- © The Author(s), 2025. Published by Cambridge University Press
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