Forcing unbalanced complete bipartite minors

@article{Khn2005ForcingUC,
  title={Forcing unbalanced complete bipartite minors},
  author={Daniela K{\"u}hn and Deryk Osthus},
  journal={Eur. J. Comb.},
  year={2005},
  volume={26},
  pages={75-81}
}
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math.uni-hamburg.de
24 Citations
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24 Citations

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TLDR
This work strengthens (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an arbitrary graph as a minor when $H$ is a sparse graph with many high degree vertices.
Cycles of Given Size in a Dense Graph
TLDR
It is shown that every graph with average degree at least $\frac{4}{3}kr$ contains $k$ vertex disjoint cycles, each of order at least $r$ as long as $k \geq 6$.
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References

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Hadwiger's Conjecture is True for Almost Every Graph
The extremal function for unbalanced bipartite minors
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.
The Extremal Function for Complete Minors
TLDR
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.
An improved linear edge bound for graph linkages
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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 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 Sufficient Degree Conditions for a Graph to be $k$-linked
TLDR
The bound allows the Thomas–Wollan proof slightly to be modified slightly to show that every $2k-connected graph with average degree at least $12k$ is $k$-linked.
The Extremal Function For Noncomplete Minors
TLDR
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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