Random Regular Graphs of Non-Constant Degree: Connectivity and Hamiltonicity

Random Regular Graphs Of Non-Constant Degree: Connectivity And Hamiltonicity

Let Gr denote a graph chosen uniformly at random from the set of r-regular graphs with vertex set f1; 2; : : : ; ng where 3 r c0n for some small constant c0. We prove that with probability tending to 1 as n ! 1, Gr is r-connected and Hamiltonian.

On random regular graphs with non-constant degree

We study random r-regular labelled graphs with n vertices. For r = 0(n ) we give the asymptotic number of them and show that they are almost all r-connected and hamiltonian. Proofs are based on the analysis of a simple algorithm for finding "almost" random regular graphs. (Diversity Libraries §

Random regular graphs of non- onstant degree

Random regular graphs of non-constant degree: distribution of edges and applications

In this work we analyze the distribution of the number of edges spanned by a single set and between two disjoint sets of vertices in the random regular graph model Gn,d in the range d = o( √ n). We show it to be very similar to that of the binomial random graph model, G(n, p) with p = d n . Some graph properties are known to exist solely based on the edge distribution of the graph. We demonstra...

Random Regular Graphs Of Non-Constant Degree: Independence And Chromatic Number

Let r(= r(n))!1 with 3 r n1 for an arbitrarily small constant > 0, and let Gr denote a graph chosen uniformly at random from the set of r-regular graphs with vertex set f1; 2; : : : ; ng. We prove that with probability tending to 1 as n!1, Gr has the following properties: the independence number of Gr is asymptotically 2n log r r and the chromatic number of Gr is asymptotically r 2 log r .