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Common pairs of graphs

Part of: Graph theory

Published online by Cambridge University Press:  30 June 2025

Natalie Behague Open the ORCID record for Natalie Behague [Opens in a new window] ,
Natasha Morrison  and
Jonathan A. Noel Open the ORCID record for Jonathan A. Noel [Opens in a new window]
Natalie Behague
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada Mathematics Institute, University of Warwick, Coventry, UK
Natasha Morrison
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada
Jonathan A. Noel*
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada
*
Corresponding author: Jonathan A. Noel; noelj@uvic.ca
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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.

Keywords

Extremal graph theorygraph limitsRamsey multiplicityquasirandomnessSidorenko’s Conjectureflag algebrasRamsey theorycommon graphs

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C55: Generalized Ramsey theory

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Type
Paper
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Combinatorics, Probability and Computing , First View , pp. 1 - 22
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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