Browse Theses

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 G k , 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 G k , n -universal graph H ý ( k, n ) of size O _k ( n ). The smallest previously known G k , n -universal graphs have size &thetas; _k ( n log n ). To prove that H ý ( k, n ) is indeed G k , n -universal, we introduce new graph-embedding techniques that exploit the fact that each Gý G k , 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 H _g k , n . For all positive &epsis;, we construct H _g k , n -universal graphs ý( k, n , &epsis;) of size O _{&epsis;, k} ( n ), where g ( x ) = x 1ý&epsis; . The smallest previously known such H _g k , n -universal graphs have size &thetas;_{&epsis;, k} ( n 2ý2&epsis; ). To prove that ý(&epsis;, k, n ) is indeed H _g k , 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 H k , n denote the family of graphs on n or fewer vertices, and with maximum degree k . We explicitly construct a H k , n -universal graph ý( k, n ) of size O _k ( n 2ý2/ k log 5 n ). The size of ý( k, n ) is nearly as small as possible, since Noga Alon has shown that any H k , n -universal graph must have size at least ý( n 2ý2/ k ). To prove that ý( k, n ) is H k , n -universal, we introduce completely new graph-embedding techniques which use probabilistic methods.

In Chapter 5 we close with some open problems.