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Near-optimum Universal Graphs for Graphs with Bounded Degrees

Extended Abstract
  • Noga Alon
  • Michael Capalbo
  • Yoshiharu Kohayakawa
  • Vojtěch Rödl
  • Andrzej Ruciński
  • Endre Szemerédi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2129)

Abstract

Let \( \mathcal{H} \) be a family of graphs. We say that G is \( \mathcal{H} \)-universal if, for each H\( \mathcal{H} \), the graph G contains a subgraph isomorphic to H. Let \( \mathcal{H} \) (k, n) denote the family of graphs on n vertices with maximum degree at most k. For each fixed k and each n sufficiently large, we explicitly construct an \( \mathcal{H} \) (k, n)-universal graph Γ(k, n) with O(n 2 - 2/k (log n)1+8/k ) edges. This is optimal up to a small polylogarithmic factor, as Ω(n 2-2/k ) is a lower bound for the number of edges in any such graph.

En route, we use the probabilistic method in a rather unusual way. After presenting a deterministic construction of the graph Γ(k, n) we prove, using a probabilistic argument, that Γ(k, n) is \( \mathcal{H} \)(k, n)-universal. So we use the probabilistic method to prove that an explicit construction satisfies certain properties, rather than showing the existence of a construction that satisfies these properties.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Noga Alon
    • 1
  • Michael Capalbo
    • 2
  • Yoshiharu Kohayakawa
    • 3
  • Vojtěch Rödl
    • 4
  • Andrzej Ruciński
    • 5
  • Endre Szemerédi
    • 6
  1. 1.Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of Mathematical SciencesThe Johns Hopkins UniversityBaltimore
  3. 3.Instituto de Matemática e EstatýsticaUniversidade de São PauloBrazil
  4. 4.Department of Mathematics and Computer ScienceEmory UniversityAtlanta
  5. 5.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland
  6. 6.Department of Computer ScienceRutgers University

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