• Corpus ID: 239009588

The extremal function for structured sparse minors

  title={The extremal function for structured sparse minors},
  author={Matthew Wales},
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 pairs of parts have many more edges than others — for instance a complete bipartite graph with a small number of edges placed inside one class. We also show a tight matching lower bound for almost all such graphs. We apply these results to show c(Kft/ log t,t) = (0.638 . . .+ of (1))t √ f , for f = o… 


The Extremal Function for Complete Minors
  • A. Thomason
  • Computer Science, Mathematics
    J. Comb. Theory, Ser. B
  • 2001
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.
Forcing a sparse minor
  • B. Reed, D. Wood
  • Mathematics, Computer Science
    Combinatorics, 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$ .
A Lower Bound on the Average Degree Forcing a Minor
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.
On Ks, t-minors in graphs with given average degree
This paper confirms the conjecture of Myers and finds asymptotically (in s) exact bounds on D(K"s","t) for a fixed s and large t and proves that for 'balanced'H random graphs provide extremal examples and determine the extremal function.
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 extremal function for graph minors
For a graph $H$, let $c(H)=\inf\{c\,:\,e(G)\geq c|G| \mbox{ implies } G\succ H\,\}$, where $G\succ H$ means that $H$ is a minor of $G$. We show that if $H$ has average degree $d$, then $$ c(H)\le
On Ks, t-minors in graphs with given average degree, II
It is shown how to adapt the argument of the previous paper to prove that if t/ log2 t ≥ 1000s, then every graph G with average degree at least t + 8s log2 s has a K ∗ s,t minor.
Forcing unbalanced complete bipartite minors
It is proved that for every 0 < e < 10-16 there exists a number t0 = t0(e) such that for all integers t ≥ t0 and s ≤ e7t/log t every graph of average degree at least (1 + e)t contains a Ks + K-t minor.
The Extremal Function For Noncomplete Minors
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.
The Probabilistic Method
A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.