Packing vertices and edges in random regular graphs

Packing vertices and edges in random regular graphs

In this paper we consider the problem of finding large collections of vertices and edges satisfying particular separation properties in random regular graphs of degree r, for each fixed r ≥ 3. We prove both constructive lower bounds and combinatorial upper bounds on the maximal sizes of these sets. The lower bounds are proved by analysing a class of algorithms that return feasible solutions for...

Flipping edges and vertices in graphs

We study a certain random process on a graph G which is a variation of a classical voter model and is also a special case of the so-called Tsetlin library random walk. Initially each vertex of G is colored either blue or red. At each step an edge is chosen at random and both endpoints change their colors to blue with probability p and to red otherwise. This edge-flipping process corresponds to ...

Hamilton cycles containing randomly selected edges in random regular graphs

In previous papers the authors showed that almost all d-regular graphs for d ≥ 3 are hamiltonian. In the present paper this result is generalized so that a set of j oriented root edges have been randomly specified for the cycle to contain. The Hamilton cycle must be orientable to agree with all of the orientations on the j root edges. It is shown that the requisite Hamilton cycle almost surely ...

Trianguled and Weakly Triangulated Graphs: Simpliciality in Vertices and Edges

We introduce the notion of weak simpliciality, in order to extend to weakly triangulated graphs properties of triangulated graphs, using Hayward’s notion that a vertex in a triangulated graph behaves as an edge in a weakly triangulated graph. In particular, we use our definition of weak simplicial edge elimination ordering to bound the number of minimal separators to n+m, and derive an efficien...

Triangulated and Weakly Triangulated Graphs: Simpliciality in Vertices and Edges