A Lower Bound on the Average Degree Forcing a Minor

Norin, Sergey; Reed, Bruce; Thomason, Andrew; Wood, David R.

doi:10.37236/8847

Cited by 6 publications

(7 citation statements)

References 31 publications

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“…0, and its leading coefficient is non-negative, also by (24). Thus (26) holds for all x ∈ R, as desired.…”

Section: From Graphs To Weight Vectorsmentioning

confidence: 74%

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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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“…0, and its leading coefficient is non-negative, also by (24). Thus (26) holds for all x ∈ R, as desired.…”

Section: From Graphs To Weight Vectorsmentioning

confidence: 74%

“…Norin, Reed, Thomason and Wood [26] recently showed that (2) is also tight for almost all graphs of constant density; that is, for almost all graphs H with d(H) = d, This paper investigates graph classes for which the extremal function behaves qualitatively differently, such as for the following concrete families:…”

Section: Introductionmentioning

confidence: 99%

Extremal functions for sparse minors

Hendrey,

Norin,

Wood

2021

Preprint

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“…with t vertices and average degree d sufficiently large. The following lower bound, true for almost all such graphs, was recently proven by Norin, Reed, Thomason and Wood [9]. They claim this result only for integer d, though this is not a necessary limitation of their proof.…”

Section: Myers and Thomasonmentioning

confidence: 90%

“…For general graphs, the extremal function is known for almost all graphs with large fixed average degree and any given number of vertices, with a lower bound due to Norin, Reed, Thomason and Wood [9] being matched by an upper bound by Thomason and the author [14]. In this paper, we generalise to two new cases.…”

Section: Introductionmentioning

confidence: 99%

The extremal function for structured sparse minors

Wales

2021

Preprint

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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(K f t/ log t,t ) = (0.638 . . . + o f (1))t √ f , for f = o(log t) = ω(1).

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“…Moreover, the argument of [11] indicates that equality holds for almost all such H . Similar remarks could be made regarding multipartite graphs.…”

Section: Further Extensionsmentioning

confidence: 99%