Maximum degree in minor-closed classes of graphs

Giménez, Omer; Mitsche, Dieter; Noy, Marc

doi:10.1016/j.ejc.2016.02.001

Cited by 5 publications

(11 citation statements)

References 15 publications

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“…A simple computation then yields: Before proving Proposition 8.3 we identify the unlabeled blocks of this class. This result (among extensions to Forb(C 6 ) and Forb(C 7 )) was given by Giménez, Mitsche and Noy [24].…”

Section: Forb(csupporting

confidence: 57%

Scaling limits of random graphs from subcritical classes

Παναγιώτου¹,

Stufler²,

Weller³

2016

Ann. Probab.

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Abstract. We study the uniform random graph Cn with n vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph Cn/ √ n converges to the Brownian Continuum Random Tree Te multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter D(Cn) and height H(C • n ) of the rooted random graph C • n . We give analytic expressions for the scaling factor of several classes, including for example the prominent class of outerplanar graphs. Our methods also enable us to study first passage percolation on Cn, where we show the convergence to Te under an appropriate rescaling.

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Section: Forb(csupporting

confidence: 57%

Scaling limits of random graphs from subcritical classes

Παναγιώτου¹,

Stufler²,

Weller³

2016

Ann. Probab.

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“…The right-hand side is very similar to the upper bound in (17). There are only two differences between these upper bounds.…”

supporting

confidence: 54%

“…The maximal degree of random combinatorial graphs has been extensively investigated, see e.g. [5,9,15,16,17,23,24]. Much less has been done when the vertices are given by a point process and the edges built according to geometric constraints.…”

Section: Introductionmentioning

confidence: 99%

The maximal degree in a Poisson–Delaunay graph

Bonnet¹,

Chenavier²

2020

Bernoulli

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We investigate the maximal degree in a Poisson-Delaunay graph in R d , d ≥ 2, over all nodes in the window Wρ := ρ 1/d [0, 1] d as ρ goes to infinity. The exact order of this maximum is provided in any dimension. In the particular setting d = 2, we show that this quantity is concentrated on two consecutive integers with high probability. An extension of this result is discussed when d ≥ 3.We summarize here the notation used throughout the text. General notationWe denote by N = {1, 2, . . .} and R + = [0, ∞) the sets of positive integers and nonnegative numbers, respectively. The d-dimensional Euclidean space R d is endowed with the Euclidean norm · and with its d-dimensional Lebesgue measure V d (·). We denote by B d + the set of Borel sets B ⊂ R d such that 0 < V d (B) < ∞. The unit sphere with dimension d − 1 is denoted by S d−1 .

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“…For graphs in this class, each block is a vertex or an edge or a triangle. Thus, for R n ∈ u Ex(C 4 ), by (3) we have ∆(R n ) ≤ (2 + o(1)) log n/ log log n whp, as in [11] Lemma 10. This inequality is tight, and we have ∆(R n ) log log n/ log n → 2 in probability as n → ∞.…”

Section: Introductionmentioning

confidence: 69%

“…Suppose that our block-stable class is the class of all series-parallel graphs or another 'subcritical' graph class, or it is the class of planar graphs, or another such class where we know the corresponding generating functions suitably well. In such cases, we may be able to deduce precise asymptotic results, for example about vertex degrees or the numbers and sizes of blocks, by using analytic techniques or by analysing Boltzmann samplers: see for example [2], [6], [7], [8], [10], [11], [12], [13], [14], [22], [23], and for an authoritative recent overview of related work on random planar graphs and beyond see the article [21] by Marc Noy.…”

Section: Introductionmentioning

confidence: 99%