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Universal and unavoidable graphs

Part of: Graph theory Extremal combinatorics

Published online by Cambridge University Press:  15 April 2021

Matija Bucić ,
Nemanja Draganić  and
Benny Sudakov
Matija Bucić
Affiliation:
Department of Mathematics, ETH, Zürich, Switzerland
Nemanja Draganić
Affiliation:
Department of Mathematics, ETH, Zürich, Switzerland
Benny Sudakov*
Affiliation:
Department of Mathematics, ETH, Zürich, Switzerland
*
*Corresponding author. Email: benjamin.sudakov@math.ethz.ch
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Abstract

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The Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C60: Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 05D40: Probabilistic methods 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 6 , November 2021 , pp. 942 - 955
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

†

Research supported in part by SNSF grant 200021_196965.

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