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The intersection of a random geometric graph with an Erdős–Rényi graph.

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Title: The intersection of a random geometric graph with an Erdős–Rényi graph.
Authors: Bennett, Patrick1 (AUTHOR), Frieze, Alan1,2 (AUTHOR) alan@random.math.cmu.edu, Pegden, Wesley1,2 (AUTHOR)
Source: Discrete Applied Mathematics. Jun2026, Vol. 386, p16-24. 9p.
Subjects: Random graphs, Intersection graph theory, Independent sets, Mathematical connectedness, Hamiltonian graph theory, Graph coloring
Abstract: We study the intersection of a random geometric graph with an Erdős–Rényi graph. Specifically, we generate the random geometric graph G (n , r) by choosing n points uniformly at random from D = [ 0 , 1 ] 2 and joining any two points whose Euclidean distance is at most r. We let G (n , p) be the classical Erdős–Rényi graph, i.e. it has n vertices and every pair of vertices is adjacent with probability p independently. In this note we study G (n , r , p) ≔ G (n , r) ∩ G (n , p). One way to think of this graph is that we take G (n , r) and then randomly delete edges with probability 1 − p independently. We consider the clique number, independence number, connectivity, Hamiltonicity, chromatic number, and diameter of this graph where both p (n) → 0 and r (n) → 0 ; the same model was studied by Kahle et al. (2023) for r (n) → 0 but p fixed. [ABSTRACT FROM AUTHOR]
Copyright of Discrete Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 192967402
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  Data: The intersection of a random geometric graph with an Erdős–Rényi graph.
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  Data: <searchLink fieldCode="AR" term="%22Bennett%2C+Patrick%22">Bennett, Patrick</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Frieze%2C+Alan%22">Frieze, Alan</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> alan@random.math.cmu.edu</i><br /><searchLink fieldCode="AR" term="%22Pegden%2C+Wesley%22">Pegden, Wesley</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Discrete+Applied+Mathematics%22">Discrete Applied Mathematics</searchLink>. Jun2026, Vol. 386, p16-24. 9p.
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  Data: <searchLink fieldCode="DE" term="%22Random+graphs%22">Random graphs</searchLink><br /><searchLink fieldCode="DE" term="%22Intersection+graph+theory%22">Intersection graph theory</searchLink><br /><searchLink fieldCode="DE" term="%22Independent+sets%22">Independent sets</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+connectedness%22">Mathematical connectedness</searchLink><br /><searchLink fieldCode="DE" term="%22Hamiltonian+graph+theory%22">Hamiltonian graph theory</searchLink><br /><searchLink fieldCode="DE" term="%22Graph+coloring%22">Graph coloring</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: We study the intersection of a random geometric graph with an Erdős–Rényi graph. Specifically, we generate the random geometric graph G (n , r) by choosing n points uniformly at random from D = [ 0 , 1 ] 2 and joining any two points whose Euclidean distance is at most r. We let G (n , p) be the classical Erdős–Rényi graph, i.e. it has n vertices and every pair of vertices is adjacent with probability p independently. In this note we study G (n , r , p) ≔ G (n , r) ∩ G (n , p). One way to think of this graph is that we take G (n , r) and then randomly delete edges with probability 1 − p independently. We consider the clique number, independence number, connectivity, Hamiltonicity, chromatic number, and diameter of this graph where both p (n) → 0 and r (n) → 0 ; the same model was studied by Kahle et al. (2023) for r (n) → 0 but p fixed. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Discrete Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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      – Type: doi
        Value: 10.1016/j.dam.2026.01.034
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      – Code: eng
        Text: English
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        PageCount: 9
        StartPage: 16
    Subjects:
      – SubjectFull: Random graphs
        Type: general
      – SubjectFull: Intersection graph theory
        Type: general
      – SubjectFull: Independent sets
        Type: general
      – SubjectFull: Mathematical connectedness
        Type: general
      – SubjectFull: Hamiltonian graph theory
        Type: general
      – SubjectFull: Graph coloring
        Type: general
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      – TitleFull: The intersection of a random geometric graph with an Erdős–Rényi graph.
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            NameFull: Bennett, Patrick
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            NameFull: Frieze, Alan
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            NameFull: Pegden, Wesley
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            – D: 15
              M: 06
              Text: Jun2026
              Type: published
              Y: 2026
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