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Subgraph densities in a surface

Part of: Graph theory

Published online by Cambridge University Press:  11 January 2022

Tony Huynh ,
Gwenaël Joret  and
David R. Wood
Tony Huynh
Affiliation:
School of Mathematics, Monash University, Melbourne, Australia
Gwenaël Joret
Affiliation:
Département d’Informatique, Université libre de Bruxelles, Brussels, Belgium
David R. Wood*
Affiliation:
School of Mathematics, Monash University, Melbourne, Australia
*
*Corresponding author. Email: david.wood@monash.edu
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Abstract

Given a fixed graph H that embeds in a surface $\Sigma$ , what is the maximum number of copies of H in an n-vertex graph G that embeds in $\Sigma$ ? We show that the answer is $\Theta(n^{f(H)})$ , where f(H) is a graph invariant called the ‘flap-number’ of H, which is independent of $\Sigma$ . This simultaneously answers two open problems posed by Eppstein ((1993) J. Graph Theory17(3) 409–416.). The same proof also answers the question for minor-closed classes. That is, if H is a $K_{3,t}$ minor-free graph, then the maximum number of copies of H in an n-vertex $K_{3,t}$ minor-free graph G is $\Theta(n^{f'(H)})$ , where f′(H) is a graph invariant closely related to the flap-number of H. Finally, when H is a complete graph we give more precise answers.

Keywords

graphgraph embeddingsurfacehomomorphism inequality

MSC classification

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 5 , September 2022 , pp. 812 - 839
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

All three authors are supported by the Australian Research Council. G. Joret is supported by an ARC grant from the Wallonia-Brussels Federation of Belgium and a CDR grant from the National Fund for Scientific Research (FNRS).

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