# The Extremal Function For Noncomplete Minors

@article{Myers2005TheEF,
title={The Extremal Function For Noncomplete Minors},
author={Joseph Samuel Myers and Andrew Thomason},
journal={Combinatorica},
year={2005},
volume={25},
pages={725-753}
}
• Published 1 December 2005
• Mathematics
• Combinatorica
We investigate the maximum number of edges that a graph G can have if it does not contain a given graph H as a minor (subcontraction). Let $$c{\left( H \right)} = \inf {\left\{ {c:e{\left( G \right)} \geqslant c{\left| G \right|}{\text{implies}}{\kern 1pt} {\kern 1pt} G \succ H} \right\}}.$$We define a parameter γ(H) of the graph H and show that, if H has t vertices, then $$c{\left( H \right)} = {\left( {\alpha \gamma {\left( H \right)} + o{\left( 1 \right)}} \right)}t{\sqrt {\log {\kern 1pt… On the extremal function for graph minors • Mathematics, Computer Science Journal of Graph Theory • 2022 It is shown that if H has average degree d, then c(H)le (0.319\ldots+o_d(1)|H|\sqrt{\log d}$$ is an explicitly defined constant, which matches a corresponding lower bound shown to hold for almost all such graphs by Norin, Reed, Wood and the first author.
Asymptotic density of graphs excluding disconnected minors
• Mathematics
J. Comb. Theory, Ser. B
• 2021
Minors in Expanding Graphs
• Mathematics
• 2007
Abstract.We propose a unifying framework for studying extremal problems related to graph minors. This framework relates the existence of a large minor in a given graph to its expansion properties. We
The extremal function for disconnected minors
• Mathematics
J. Comb. Theory, Ser. B
• 2017
A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary
• Mathematics, Computer Science
SODA
• 2020
Several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement are combined.
Densities of Minor-Closed Graph Families
By analyzing density-minimal graphs of low densities, this work finds all limiting densities up to the first two cluster points of the set of limiting density, $1$ and $3/2$.
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$. Extremal Density for Sparse Minors and Subdivisions • Mathematics • 2020 We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several A Lower Bound on the Average Degree Forcing a Minor • Mathematics, Computer Science Electron. J. Comb. • 2020 It is shown that for sufficiently large graphs and for t ≥ d, there is a graph G such that almost every graph H with t vertices and average degree$H$is not a minor of$G$, where$\lambda=0.63817\dots$is an explicitly defined constant. The extremal function for structured sparse minors Let c(H) be the smallest value for which e(G)/|G| > c(H) implies H is a minor of G. We show a new upper bound on c(H), which improves previous bounds for graphs with a vertex partition where some ## References SHOWING 1-10 OF 22 REFERENCES Graph Minors: XV. Giant Steps • Mathematics J. Comb. Theory, Ser. B • 1996 Abstract Let G be a graph with a subgraph H drawn with high representativity on a surface Σ . When can the drawing of H be extended “up to 3-separations” to a drawing of G in Σ if we permit a bounded Spanning Subgraphs of Random Graphs Let Gp be a random graph on 2 d vertices where edges are selected independently with a xed probability p > 1 4 , and let H be the d-dimensional hypercube Q d. We answer a question of Bollobb as by The Extremal Function for Complete Minors Let c(t) be the minimum number c such that every graph G with e(G)?c|G| contracts to a complete graph Kt where ?=0.319... is an explicit constant. Graphs without Large Complete Minors are Quasi-Random • J. Myers • Mathematics Combinatorics, Probability and Computing • 2002 It is shown that, if a graph G of order n and density p has no complete minor larger than would be found in a random graph G(n, p), then G is quasi-random, provided either p > 0.45631 … or κ(G) [ges ] n( log log log n)/(log log n), where 0.45731 … is an explicit constant. A LIMIT THEOREM IN GRAPH THEORY • Mathematics • 1966 In this paper G(n ; I) will denote a graph of n vertices and l edges, K„ will denote the complete graph of p vertices G (p ; (PA and K,(p i , . . ., p,) will denote the rchromatic graph with p i Highly linked graphs • Mathematics Comb. • 1996 It is shown here that k (G)≥22k will do and that a graphG isk-linked provided its vertex connectivityk(G) exceeds 10k\sqrt {\log _2 k}$\$ .
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
Graph Minors .XIII. The Disjoint Paths Problem
• Mathematics
J. Comb. Theory, Ser. B
• 1995
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.