Exact colorations of graphs and digraphs

A coloration is an exact regular coloration if whenever two vertices are colored the same they have identically colored neighborhoods. For example, if one of the two vertices that are colored the same is connected to three yellow vertices, two white and red, then the other vertex is as well. Exact regular colorations have been discussed informally in the social network literature. However they have been part of the mathematical literature for some time, though in a different format. We explore this concept in terms of social networks and illustrate some important results taken from the mathematical literature. In addition we show how the concept can be extended to ecological and perfect colorations, and discuss how the CATREGE algorithm can be extended to find the maximal exact regular coloration of a graph.

A generational scheme for partitioning graphs

Graph partitioning divides a graph into several pieces by cutting edges. Very effective heuristic partitioning algorithms have been developed which run in real-time, but it is unknown how good the partitions are since the problem is, in general, NP-complete. This paper reports an evolutionary search algorithm for finding benchmark partitions. Distinctive features are the transmission and modification of whole subdomains (the partitioned units) that act as genes, and the use of a multilevel heuristic algorithm to effect the crossover and mutations. Its effectiveness is demonstrated by improvements on previously established benchmarks.

On Hamilton cycles in locally connected graphs with vertex degree constraints

It is shown that every connected, locally connected graph with the maximum vertex degree Δ(G)=5 and the minimum vertex degree δ(G)3 is fully cycle extendable. For Δ(G)4, all connected, locally connected graphs, including infinite ones, are explicitly described. The Hamilton Cycle problem for locally connected graphs with Δ(G)7 is shown to be NP-complete

Exploring the conditional performance of UK unit trusts

We evaluate the conditional performance of U.K. equity unit trusts using the approach of Lynch and Wachter (2007, 2008) relative to three conditional linear factor models. We find significant time variation in the conditional performance of some trust portfolios and individual trusts using the lag term spread as the information variable. The conditional performance of the trusts is countercyclical and larger trusts have more countercyclical performance than smaller trusts within certain investment sectors. These patterns in conditional trust performance cannot be fully explained by the underlying securities that the trusts hold.

Distress in children with learning disabilities at a respite unit: perspective on their experiences

Numerous studies have investigated the benefits of respite to families with a disabled child. Far fewer have examined the effects on the child and none have systematically compared information about this from different sources. Reports of behavioural reactions and views on distress were gathered from parents, teachers and respite staff. Children were also asked for their views. Over half the children (54%) were reported to show medium or strong negative reactions lasting for 1 or more days by a parent or teacher. Reported reactions varied widely between home and school and no concordance was found between parents, teachers and respite staff groups regarding distress. Some children's views differed from those of their parent or teacher. The findings highlight the extent of differences in perspectives and suggest the need for greater awareness of the possible distress to children attending respite. This is discussed in relation to factors such as the potential conflict of interests between parents and children, communication and behavioural difficulties, and the context in which the child is observed.