The Extremal Function for Complete Minors

@article{Thomason2001TheEF,
  title={The Extremal Function for Complete Minors},
  author={Andrew Thomason},
  journal={J. Comb. Theory, Ser. B},
  year={2001},
  volume={81},
  pages={318-338}
}
  • A. Thomason
  • Published 1 March 2001
  • Mathematics
  • J. Comb. Theory, Ser. B
Let c(t) be the minimum number c such that every graph G with e(G)?c|G| contracts to a complete graph Kt. We show thatc(t)=(?+o(1))tlogtwhere ?=0.319... is an explicit constant. Random graphs are extremal graphs. 
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References

SHOWING 1-10 OF 36 REFERENCES
Graph Minors .XIII. The Disjoint Paths Problem
TLDR
An algorithm, which for fixed k ≥ 0 has running time O (| V(G) | 3 ), to solve the following problem: given a graph G and k pairs of vertices of G, decide if there are k mutually vertex-disjoint paths of G joining the pairs.
Hadwiger's Conjecture is True for Almost Every Graph
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 contractability of an acute oriented graph in K
Graphs without Large Complete Minors are Quasi-Random
  • J. Myers
  • Mathematics
    Combinatorics, Probability and Computing
  • 2002
TLDR
It is shown that, if a graph G of order n and density p has no complete minor larger than would be found in a random graph G(n, p), then G is quasi-random, provided either p > 0.45631 … or κ(G) [ges ] n( log log log n)/(log log n), where 0.45731 … is an explicit constant.
Pseudo-Random Graphs
Contractions to k8
It is proved that the maximal number of edges in a graph with n ≧ 8 vertices that is not contractible to K8 is 6n − 21, unless 5 divides n, and the only graphs with n = 5m vertices and more than 6n −…
Lower bound of the hadwiger number of graphs by their average degree
The aim of this paper is to show that the minimum Hadwiger number of graphs with average degreek isO(k/√logk). Specially, it follows that Hadwiger’s conjecture is true for almost all graphs withn…
Highly linked graphs
TLDR
It is shown here that k (G)≥22k will do and that a graphG isk-linked provided its vertex connectivityk(G) exceeds 10k\sqrt {\log _2 k}$$ .
An extremal function for digraph subcontraction
TLDR
It is shown that the maximum size of a digraph which has no subcontraction to the complete digraph DK, of order p, holds if p is sufficiently large, and the function d(p) differs by only a constant factor from the corresponding function for undirected graphs.
...
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