On Ks, t-minors in graphs with given average degree, II

@article{Kostochka2012OnKT,
  title={On Ks, t-minors in graphs with given average degree, II},
  author={Alexandr V. Kostochka and Noah Prince},
  journal={Discret. Math.},
  year={2012},
  volume={312},
  pages={3517-3522}
}

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References

SHOWING 1-10 OF 23 REFERENCES
On Ks,t minors in (s+t)‐chromatic graphs
TLDR
It is proved that for each fixed s and sufficiently large t, every graph with chromatic number s+t has a K∗ s,t minor.
On K s,t minors in (s+t)-chromatic graphs
Let K *s, t denote the graph obtained from the complete graph Ks+t by deleting the edges of some Kt-subgraph. We prove that for each fixed s and sufficiently large t, every graph with chromatic
On Ks, t-minors in graphs with given average degree
The edge-density for K 2 , t minors
Let H be a graph. If G is an n-vertex simple graph that does not contain H as a minor, what is the maximum number of edges that G can have? This is at most linear in n, but the exact expression is
The edge-density for K2, t 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.
Dense graphs have K3, t minors
The extremal function for unbalanced bipartite minors
Forcing unbalanced complete bipartite minors
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.
...
1
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3
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