The Extremal Function For Noncomplete Minors

@article{Myers2005TheEF,
  title={The Extremal Function For Noncomplete Minors},
  author={Joseph Samuel Myers and Andrew Thomason},
  journal={Combinatorica},
  year={2005},
  volume={25},
  pages={725-753}
}
We investigate the maximum number of edges that a graph G can have if it does not contain a given graph H as a minor (subcontraction). Let $$ c{\left( H \right)} = \inf {\left\{ {c:e{\left( G \right)} \geqslant c{\left| G \right|}{\text{implies}}{\kern 1pt} {\kern 1pt} G \succ H} \right\}}. $$We define a parameter γ(H) of the graph H and show that, if H has t vertices, then $$ c{\left( H \right)} = {\left( {\alpha \gamma {\left( H \right)} + o{\left( 1 \right)}} \right)}t{\sqrt {\log {\kern 1pt… 
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37 Citations

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TLDR
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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Forcing a sparse minor
  • B. Reed, D. Wood
  • Mathematics
    Combinatorics, Probability and Computing
  • 2015
TLDR
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$ .
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We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several…
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TLDR
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.
The extremal function for structured sparse minors
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…
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