Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count

Eppstein, David

doi:10.1007/978-3-319-73915-1_39

Cited by 5 publications

(3 citation statements)

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“…In geometric graph theory, penny graphs are contact graphs of unit circles in the plane. Finite penny graphs are extensively studied in the literature, for example, [4, 9, 12–14, 20, 24–28]. In this paper, we study discrete harmonic functions of polynomial growth on infinite penny graphs.…”

Section: Introductionmentioning

confidence: 99%

Harmonic functions of polynomial growth on infinite penny graphs

Hua

2022

Journal of London Math Soc

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Section: Introductionmentioning

confidence: 99%

Harmonic functions of polynomial growth on infinite penny graphs

Hua

2022

Journal of London Math Soc

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“…They are the graphs formed by arranging pennies in a non-overlapping way on the plane, making a vertex for each penny, and making an edge for each two pennies that touch. Finite penny graphs are extensively studied in the literature, to cite a few [Har74, Pol85, Kup94, PA95, PT96, Csi98, PR00, HK01, Swa09, CFFP11,Epp18]. In this paper, we study infinite penny graphs and the function theory defined on them.…”

Section: Introductionmentioning

confidence: 99%

Discrete harmonic functions on infinite penny graphs

Hua

2020

Preprint

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In this paper, we study discrete harmonic functions on infinite penny graphs. For an infinite penny graph with bounded facial degree, we prove that the volume doubling property and the Poincaré inequality hold, which yields the Harnack inequality for positive harmonic functions. Moreover, we prove that the space of polynomial growth harmonic functions, or ancient solutions of the heat equation, with bounded growth rate has finite dimensional property.

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“…If all coins have the same size, the represented graphs are called penny graphs. These graphs have been studied extensively, too [4,8,11]. For example, they are NP-hard to recognize [3,7].…”

Section: Introductionmentioning

confidence: 99%

On Arrangements of Orthogonal Circles

Chaplick

Förster

Kryven

et al. 2019

Lecture Notes in Computer Science

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In this paper, we study arrangements of orthogonal circles, that is, arrangements of circles where every pair of circles must either be disjoint or intersect at a right angle. Using geometric arguments, we show that such arrangements have only a linear number of faces. This implies that orthogonal circle intersection graphs have only a linear number of edges. When we restrict ourselves to orthogonal unit circles, the resulting class of intersection graphs is a subclass of penny graphs (that is, contact graphs of unit circles). We show that, similarly to penny graphs, it is NPhard to recognize orthogonal unit circle intersection graphs.

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Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count

Abstract: We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter D = Ω( √ n), and at most min 2n − Ω( √ n), 2n − D − 2 edges.

Cited by 5 publications

References 24 publications

Harmonic functions of polynomial growth on infinite penny graphs

Harmonic functions of polynomial growth on infinite penny graphs

Discrete harmonic functions on infinite penny graphs

On Arrangements of Orthogonal Circles

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