# Stefanie Gerke

• Discrete Mathematics
• 2009
A uniform random intersection graph G(n,m, k) is a random graph constructed as follows. Label each of n nodes by a randomly chosen set of k distinct colours taken from some finite set of possible colours of size m. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless(More)
• J. Comb. Theory, Ser. B
• 2001
We are interested in colouring a graph G=(V, E) together with an integral weight or demand vector x=(xv : v # V) in such a way that xv colours are assigned to each node v, adjacent nodes are coloured with disjoint sets of colours, and we use as few colours as possible. Such problems arise in the design of cellular communication systems, when radio channels(More)
• J. Comb. Theory, Ser. B
• 2001
The imperfection ratio is a graph invariant which indicates how good a lower bound the weighted clique number gives on the weighted chromatic number, in the limit as weights get large. Its introduction was motivated by investigations of the radio channel assignment problem, where one has to assign channels to transmitters and the demands for channels at(More)
• J. Comb. Theory, Ser. B
• 2008
A non-empty class A of labelled graphs is weakly addable if for each graph G ∈ A and any two distinct components of G, any graph that can be obtain by adding an edge between the two components is also in A. For a weakly addable graph class A, we consider a random element Rn chosen uniformly from the set of all graph in A on the vertex set {1, . . . , n}.(More)
Let P(n, m) be the class of simple labelled planar graphs with n nodes and m edges, and let Rn,q be a graph drawn uniformly at random from P(n, bqnc). We show properties that hold with high probability (w.h.p.) for Rn,q when 1 < q < 3. For example, we show that Rn,q contains w.h.p. linearly many nodes of each given degree and linearly many node disjoint(More)
A non-empty class A of labelled graphs is weakly addable if for each graph G ∈ A and any two distinct components of G, any graph that can be obtain by adding an edge between the two components is also in A. For a weakly addable graph class A, we consider a random element Rn chosen uniformly from the set of all graph in A on the vertex set {1, . . . , n}.(More)
In this paper we investigate the behaviour of subgraphs of sparse ε-regular bipartite graphs G = (V1 ∪ V2, E) with vanishing density d that are induced by small subsets of vertices. In particular, we show that, with overwhelming probability, a random set S ⊆ V1 of size s 1/d contains a subset S′ with |S′| ≥ (1 − ε′)|S| that induces together with V2 an(More)