Common Pairs of Graphs

TL;DR

This paper extends the concept of common graphs to pairs of graphs in an asymmetric setting, analyzing how certain linear combinations of their densities are minimized in random colourings, and presents new results and open problems.

Contribution

It introduces the asymmetric notion of common pairs of graphs, extending existing results and providing novel findings in the classical and asymmetric contexts.

Findings

Extended common graph results to asymmetric pairs.

Identified new examples of common graphs.

Proposed open problems in the field.

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 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.