# 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}
}
• Published 2005
• Mathematics
• Eur. J. Comb.
Disproof of a Conjecture by Woodall
In 2001, in a survey article [26] about list coloring, Woodall conjectured that for every pair of integers s, t ≥ 1, all graphs without a Ks,t-minor are (s + t − 1)-choosable. In this note we refute
Forcing a sparse minor
• Mathematics
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$. Linear connectivity forces large complete bipartite minors • Mathematics J. Comb. Theory, Ser. B • 2009 The extremal function for Petersen minors • Mathematics J. Comb. Theory, Ser. B • 2018 Small minors in dense graphs • Mathematics Eur. J. Comb. • 2012 Average Degree Conditions Forcing a Minor • Mathematics Electron. J. Comb. • 2016 This work strengthens (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an arbitrary graph as a minor when$H$is a sparse graph with many high degree vertices. Cycles of Given Size in a Dense Graph • Mathematics SIAM J. Discret. Math. • 2015 It is shown that every graph with average degree at least$\frac{4}{3}kr$contains$k$vertex disjoint cycles, each of order at least$r$as long as$k \geq 6$. ## References SHOWING 1-10 OF 16 REFERENCES Graphs without Large Complete Minors are Quasi-Random • J. Myers • Mathematics Combinatorics, Probability and Computing • 2002 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 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. 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 • Mathematics Comb. • 1996 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 • Mathematics Combinatorics, Probability and Computing • 2006 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
• Mathematics
Comb.
• 2005
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.