Stefanie Gerke

Graph Imperfection

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 must be assigned to transmitters to satisfy demand and avoid interference. We are particularly interested in the ratio of chromatic number to clique number when some weights are large. We introduce a relevant new graph invariant, the “imperfection ratio” imp(G) of a graph G, present alternative equivalent descriptions, and show some basic properties. For example, imp(G)=1 if and only if G is perfect, imp(G)=imp(G) where G denotes the complement of G, and imp(G)=g/(g?1) for any line graph G where g is the minimum length of an odd hole (assuming there is an odd hole).

On the Number of Edges in Random Planar Graphs

We consider random planar graphs on

Graph Imperfection II

n

Note: Generalised acyclic edge colourings of graphs with large girth

labelled nodes, and show in particular that if the graph is picked uniformly at random then the expected number of edges is at least

K 4 -free subgraphs of random graphs revisited

\frac{13}{7}n +o(n)

Algorithms for generating convex sets in acyclic digraphs

. To prove this result we give a lower bound on the size of the set of edges that can be added to a planar graph on

Nonvertex-Balanced Factors in Random Graphs

n

Colouring weighted bipartite graphs with a co-site constraint

nodes and

Surveys in Combinatorics 2013

m

Channel Assignment with Large Demands

edges while keeping it planar, and in particular we see that if