TY - GEN

T1 - Near-optimum universal graphs for graphs with bounded degrees (Extended abstract)

AU - Alon, Noga

AU - Capalbo, Michael

AU - Kohayakawa, Yoshiharu

AU - Rödl, Vojtěch

AU - Ruciński, Andrzej

AU - Szemerédi, Endre

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2001

PY - 2015

Y1 - 2015

N2 - Let H be a family of graphs. We say that G is H-universal if, for each H ∈H, the graph G contains a subgraph isomorphic to H. Let 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 H(k, n)-universal graph Γ(k, n) with O(n2−2/k(log n)1+8/k) edges. This is optimal up to a small polylogarithmic factor, as Ω(n2−2/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 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.

AB - Let H be a family of graphs. We say that G is H-universal if, for each H ∈H, the graph G contains a subgraph isomorphic to H. Let 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 H(k, n)-universal graph Γ(k, n) with O(n2−2/k(log n)1+8/k) edges. This is optimal up to a small polylogarithmic factor, as Ω(n2−2/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 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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M3 - Conference contribution

AN - SCOPUS:84923070422

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 170

EP - 180

BT - Approximation, Randomization, and Combinatorial Optimization

A2 - Trevisan, Luca

A2 - Jansen, Klaus

A2 - Goemans, Michel

A2 - Rolim, Jose D. P.

PB - Springer Verlag

T2 - 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001

Y2 - 18 August 2001 through 20 August 2001

ER -