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Local Conditions for Exponentially Many Subdivisions

Part of: Graph theory

Published online by Cambridge University Press:  28 November 2016

HONG LIU ,
MARYAM SHARIFZADEH  and
KATHERINE STADEN
HONG LIU
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 2AL, UK (e-mail: h.liu.9@warwick.ac.uk, k.l.staden@warwick.ac.uk, m.sharifzadeh@warwick.ac.uk)
MARYAM SHARIFZADEH
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 2AL, UK (e-mail: h.liu.9@warwick.ac.uk, k.l.staden@warwick.ac.uk, m.sharifzadeh@warwick.ac.uk)
KATHERINE STADEN
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 2AL, UK (e-mail: h.liu.9@warwick.ac.uk, k.l.staden@warwick.ac.uk, m.sharifzadeh@warwick.ac.uk)
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Abstract

Given a graph F, let st(F) be the number of subdivisions of F, each with a different vertex set, which one can guarantee in a graph G in which every edge lies in at least t copies of F. In 1990, Tuza asked for which graphs F and large t, one has that st(F) is exponential in a power of t. We show that, somewhat surprisingly, the only such F are complete graphs, and for every F which is not complete, st(F) is polynomial in t. Further, for a natural strengthening of the local condition above, we also characterize those F for which st(F) is exponential in a power of t.

MSC classification

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory
Secondary: 05C35: Extremal problems 05C83: Graph minors
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 3 , May 2017 , pp. 423 - 430
Copyright
Copyright © Cambridge University Press 2016 

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