# 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…

## 37 Citations

On the extremal function for graph minors

- Mathematics, Computer ScienceJournal 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.

Asymptotic density of graphs excluding disconnected minors

- MathematicsJ. Comb. Theory, Ser. B
- 2021

Minors in Expanding Graphs

- Mathematics
- 2007

Abstract.We propose a unifying framework for studying extremal problems related to graph minors. This framework relates the existence of a large minor in a given graph to its expansion properties. We…

The extremal function for disconnected minors

- MathematicsJ. Comb. Theory, Ser. B
- 2017

A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary

- Mathematics, Computer ScienceSODA
- 2020

Several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement are combined.

Densities of Minor-Closed Graph Families

- MathematicsElectron. J. Comb.
- 2010

By analyzing density-minimal graphs of low densities, this work finds all limiting densities up to the first two cluster points of the set of limiting density, $1$ and $3/2$.

Forcing a sparse minor

- MathematicsCombinatorics, Probability and Computing
- 2015

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$ .

Extremal Density for Sparse Minors and Subdivisions

- Mathematics
- 2020

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…

A Lower Bound on the Average Degree Forcing a Minor

- Mathematics, Computer ScienceElectron. 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.

The extremal function for structured sparse minors

- Mathematics
- 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…

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