• Corpus ID: 239009588

# The extremal function for structured sparse minors

@inproceedings{Wales2021TheEF,
title={The extremal function for structured sparse minors},
author={Matthew Wales},
year={2021}
}
Let c(H) be the smallest value for which e(G)/|G| > c(H) implies H is a minor of G. We show a new upper bound on c(H), which improves previous bounds for graphs with a vertex partition where some pairs of parts have many more edges than others — for instance a complete bipartite graph with a small number of edges placed inside one class. We also show a tight matching lower bound for almost all such graphs. We apply these results to show c(Kft/ log t,t) = (0.638 . . .+ of (1))t √ f , for f = o…

## References

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Let c(t) be the minimum number c such that every graph G with e(G)?c|G| contracts to a complete graph Kt where ?=0.319... is an explicit constant.
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The main result, which complements the result of Myers and Thomason, states that if H has t vertices and average degree d at least some absolute constant, then f(H) ≤ 3.895\sqrt{\ln d}\,t$. A Lower Bound on the Average Degree Forcing a Minor • Mathematics, Computer Science Electron. J. Comb. • 2020 It is shown that for sufficiently large graphs and for t ≥ d, there is a graph G such that almost every graph H with t vertices and average degree$H$is not a minor of$G$, where$\lambda=0.63817\dots$is an explicitly defined constant. An extremal function for contractions of graphs The function c ( p ) is defined for positive integers p ≥ 4 by where > denotes contraction. Random graph examples show In 1968 Mader showed that c ( p ) ≤ 8( p − 2) [log 2 ( p − 2)] and more recently On the extremal function for graph minors • Mathematics, Computer Science Journal of Graph Theory • 2022 It is shown that if$H$has average degree$d$, then c(H)le (0.319\ldots+o_d(1)|H|\sqrt{\log d}$\$ is an explicitly defined constant, which matches a corresponding lower bound shown to hold for almost all such graphs by Norin, Reed, Wood and the first author.
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The maximum number of edges that a graph G can have if it does not contain a given graph H as a minor (subcontraction) is investigated and a parameter γ(H) of the graph H is defined, equality holding for almost all H and for all regular H.
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