On the extremal function for graph minors

Thomason, Andrew; Wales, Matthew

doi:10.1002/jgt.22811

Cited by 4 publications

(4 citation statements)

References 17 publications

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“…Improving on results of [25,28], Thomason and Wales [36] recently extended the upper bound from (1.1) to general graphs, by showing that for every graph H,…”

Section: Introductionmentioning

confidence: 94%

“…Let G = G(n, 1 2 ) be the Erdős-Renyi random graph with vertex set [n], where each pair of distinct vertices are adjacent with probability 1 2 . We upper bound the probability that a fixed map µ : V (H) → P( Lemma 9.10 combined with the upper bound of Thomason and Wales [36] implies that for every ε > 0 and every graph d-regular graph G with d log 1+ε v(H), c(H) = Θ ε v(H) log d .…”

Section: General Boundsmentioning

confidence: 99%

“…Mader [22] proved that c(H) is finite for every graph H. The exact value has been determined for various small graphs H. For example, if K t is the complete graph on t 9 vertices, then c(K t ) = t − 2 (see [4,12,23,32]); and if P is the Petersen graph, then c(P) = 5 (see [11]). Here we focus on asymptotic results for classes of graphs H.…”

Section: Introductionmentioning

confidence: 99%

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Extremal functions for sparse minors

2022

AiC

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The notion of a graph minor, which generalizes graph subgraphs, is a central notion of modern graph theory. Classical results concerning graph minors include the Graph Minor Theorem and the Graph Structure Theorem, both due to Robertson and Seymour. The results concern properties of classes of graphs closed under taking minors; such graph classes include many important natural classes of graphs, e.g., the class of planar graphs and, more generally, the class of graphs embeddable in a fixed surface. The Graph Minor Theorem asserts that every class of graphs closed under taking minors has a finite list of forbidden minors. For example, Wagner’s Theorem, which claims that a graph is planar if and only if it does not contain or as a minor, is a particular case of this theorem. The Graph Structure Theorem asserts that graphs from a fixed class of graphs closed under taking minors can be decomposed in a tree-like fashion into graphs almost embeddable in a fixed surface. In particular, every graph in a class of graphs avoiding a fixed minor admits strongly sublinear separators (the Planar separator theorem of Lipton and Tarjan is a special case of this more general result). As the number of edges of every graph contained in a class of graphs closed under taking minors is linear in the number of its vertices, one can define to be the maximum possible density of a graph that does not contain a graph as a minor. This quantity has been a subject of very intensive research; for example, a long list of bounds concerning culminated with a result of Thomason in 2001, who precisely determined its asymptotic behavior. This paper provides bounds on when itself is from a class of sparse graphs. In particular, the authors prove an asymptotically tight bound on in terms of the number of vertices of and the ratio of the vertex cover and the number of vertices of graphs contained in a class of graphs with strongly sublinear separators.

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“…Improving on results of [25,28], Thomason and Wales [36] recently extended the upper bound from (1.1) to general graphs, by showing that for every graph H,…”

Section: Introductionmentioning

confidence: 94%

Section: General Boundsmentioning

confidence: 99%

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Extremal functions for sparse minors

2022

AiC

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“…implicitly using (in the language of [6]) two subsets P (the projector) to project subsets to logarithmically smaller sets, and a C (the connector) to connect small sets. However, we do not know that H ′ has suitable connectivity, and so instead use the reserved set S to serve these roles.…”

Section: Theorem 42 ([5]mentioning

confidence: 99%

Bipartite clique minors in graphs of large Hadwiger number

Wales

2021

Preprint

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The Hadwiger number h(G) is the order of the largest complete minor in G. Does sufficient Hadwiger number imply a minor with additional properties? In [2], Geelen et al showed h(G)(1 + o(1))ct √ ln t implies G has a bipartite subgraph with Hadwiger number at least t, for some explicit c ∼ 1.276 . . . . We improve this to h(G)(1 + o(1))t log 2 t, and provide a construction showing this is tight. We also derive improved bounds for the topological minor variant of this problem.

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