Asymptotic density of graphs excluding disconnected minors

Kapadia, Rohan; Norin, Sergey; Qian, Yingjie

doi:10.1016/j.jctb.2020.09.007

Cited by 3 publications

(1 citation statement)

References 22 publications

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“…Note that the assumption on regularity in the above lower bounds is necessary. For example, as shown by Kapadia, Norin and Qian [13], c(K s,t ) = Θ( st log s + s + t) for all integers t s 2. Thus the extremal function of graphs K s, s log s is linear in the number of vertices, while they have much higher density than any regular graphs with this property.…”

Section: General Boundsmentioning

confidence: 97%

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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Section: General Boundsmentioning

confidence: 97%

Extremal functions for sparse minors

2022

AiC

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Graph partitions under average degree constraint

Wang,

2024

Journal of Combinatorial Theory, Series B

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Extremal functions for sparse minors

Hendrey,

Norin,

Wood

2021

Preprint

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The extremal function c(H) of a graph H is the supremum of densities of graphs not containing H as a minor, where the density of a graph G is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005), Norin, Reed, Thomason and Wood (2020), and Thomason and Wales (2019) determined the asymptotic behaviour of c(H) for all polynomially dense graphs H, as well as almost all graphs H of constant density.We explore the asymptotic behavior of the extremal function in the regime not covered by the above results, where in addition to having constant density the graph H is in a graph class admitting strongly sublinear separators. We establish asymptotically tight bounds in many cases. For example, we prove that for every planar graph H,extending recent results of Haslegrave, Kim and Liu (2020). We also show that an asymptotically tight bound on the extremal function of graphs in minor-closed families proposed by Haslegrave, Kim and Liu ( 2020) is equivalent to a well studied open weakening of Hadwiger's conjecture.

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Asymptotic density of graphs excluding disconnected minors

Cited by 3 publications

References 22 publications

Extremal functions for sparse minors

Extremal functions for sparse minors

Graph partitions under average degree constraint

Extremal functions for sparse minors

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