Sequence variations of the 1-2-3 Conjecture and irregularity strength

Karonski, Luczak, and Thomason (2004) conjectured that, for any connected graph G on at least three vertices, there exists an edge weighting from {1,2,3} such that adjacent vertices receive different sums of incident edge weights. Bartnicki, Grytczuk, and Niwcyk (2009) made a stronger conjecture, that each edge's weight may be chosen from an arbitrary list of size 3 rather than {1,2,3}. We examine a variation of these conjectures, where each vertex is coloured with a sequence of edge weights. Such a colouring relies on an ordering of the graph's edges, and so two variations arise -- one where we may choose any ordering of the edges and one where the ordering is fixed. In the former case, we bound the list size required for any graph. In the latter, we obtain a bound on list sizes for graphs with sufficiently large minimum degree. We also extend our methods to a list variation of irregularity strength, where each vertex receives a distinct sequence of edge weights.Comment: Accepted to Discrete Mathematics and Theoretical Computer Scienc

An analysis of industry's perspective on the recent changes to Circular A-76

Acquisition research (Graduate School of Business & Public Policy)Determining whether to obtain services in-house or through commercial contracts is an important economic and strategic decision for agencies. According to Office of Management and Budget July 2003 estimates, 26 percent of the workforce from agencies being tracked under the President?s Management Agenda are engaged in commercial activities that should be available for competition. In light of the fact that the Department of Defense has achieved greater than 30 percent savings on the roughly 3,000 competitions it has conducted since 1979, there appears to be plenty of room left for harvesting savings [Ref. 1]. Not surprisingly, use of Circular A-76 is expected to grow throughout the federal government.Approved for public release; distribution is unlimited