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7 - Extremal theory of vertex or edge ordered graphs

Published online by Cambridge University Press:  17 June 2019

By
Gábor Tardos
Edited by
Allan Lo ,
Richard Mycroft ,
Guillem Perarnau  and
Andrew Treglown
Allan Lo
Affiliation:
University of Birmingham
Richard Mycroft
Affiliation:
University of Birmingham
Guillem Perarnau
Affiliation:
Universitat Politècnica de Catalunya, Barcelona
Andrew Treglown
Affiliation:
University of Birmingham
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Summary

We enrich the structure of finite simple graphs with a linear order on either the vertices or the edges. Extending the standard question of Turan-type extremal graph theory we ask for the maximal number of edges in such a vertex or edge ordered graph on n vertices that does not contain a given pattern (or several patterns) as a subgraph. The forbidden subgraph itself is also a vertex or edge ordered graph, so we forbid a certain subgraph with a specified ordering, but we allow the same underlying subgraph with a different (vertex or edge) order. This allows us to study a large number of extremal problems that are not expressible in the classical theory. In this survey we report ongoing research. For easier access, we include sketches of proofs of selected results.

Keywords

extremal combinatoricsgraph theoryordered graphsTuran theoremErdos-Stone-Simonovits theoremconvex geometric graphs
Type
Chapter
Information
Surveys in Combinatorics 2019 , pp. 221 - 236
Publisher: Cambridge University Press
Print publication year: 2019

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