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• W2912124299 abstract "Let F be a family of graphs. We say that a graph H is F -universal if each member of F is isomorphic to a subgraph of H. We construct small universal graphs for several families of bounded-degree graphs. In Chapter 2 we construct small universal graphs for families of bounded-degree planar graphs. More specifically, for all positive integers k and n, let Gk,n denote the family of planar graphs on n or fewer vertices, and with maximum degree k. For every two positive integers n and k, we construct a Gk,n -universal graph H′( k, n) of size Ok(n). The smallest previously known Gk,n -universal graphs have size tk( n log n). To prove that H ′(k, n) is indeed Gk,n -universal, we introduce new graph-embedding techniques that exploit the fact that each G∈Gk,n has a small separator function. In Chapter 3 we extend the techniques that we used in Chapter 2 to construct small universal graphs for larger families of graphs. More specifically, we say that a graph G has a function g as a 2-sector function if every subgraph of G on x vertices has a 2-sector with no more than g( x) vertices, where x is any nonnegative integer no greater than |V(G)|. For all positive integers k and n, we write the family of graphs G on n vertices or fewer that have maximum degree k, such that G has g as a 2-sector function, as Hgk, n . For all positive e, we construct Hgk, n -universal graphs Γ(k, n, e) of size Oe,k(n), where g(x) = x1−e . The smallest previously known such Hgk, n -universal graphs have size te, k(n2−2e). To prove that Γ(e, k, n) is indeed Hgk, n -universal, we refine the graph-embedding techniques introduced in Chapter 2. In Chapter 4 we construct universal graphs for the family of general bounded-degree graphs. More specifically, for all positive integers k and n, let Hk,n denote the family of graphs on n or fewer vertices, and with maximum degree k. We explicitly construct a Hk,n -universal graph Γ(k, n) of size O k(n2−2/ k log5 n). The size of Γ( k, n) is nearly as small as possible, since Noga Alon has shown that any Hk,n -universal graph must have size at least Ω(n 2−2/k). To prove that Γ( k, n) is Hk,n -universal, we introduce completely new graph-embedding techniques which use probabilistic methods. In Chapter 5 we close with some open problems." @default.
• W2912124299 created "2019-02-21" @default.
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• W2912124299 date "2001-01-01" @default.
• W2912124299 modified "2023-09-27" @default.
• W2912124299 title "Universal graphs" @default.
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