Coloring with defects

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995.Includes bibliographical references (leaves 19-21).by C.E. Jesurum.M.S

Very Cost Effective Domination in Graphs

A set S of vertices in a graph G=(V,E) is a dominating set if every vertex in V\S is adjacent to at least one vertex in S, and the minimum cardinality of a dominating set of G is the domination number of G. A vertex v in a dominating set S is said to be very cost effective if it is adjacent to more vertices in V\S than to vertices in S. A dominating set S is very cost effective if every vertex in S is very cost effective. The minimum cardinality of a very cost effective dominating set of G is the very cost effective domination number of G. We first give necessary conditions for a graph to have equal domination and very cost effective domination numbers. Then we determine an upper bound on the very cost effective domination number for trees in terms of their domination number, and characterize the trees which attain this bound. lastly, we show that no such bound exists for graphs in general, even when restricted to bipartite graphs

Low-degree graph partitioning via local search with applications to constraint satisfaction, max cut, and coloring

We present practical algorithms for constructing partitions of graphs into a fixed number of vertex-disjoint subgraphs that satisfy particular degree constraints. We use this in particular to find k-cuts of graphs of maximum degree \Delta that cut at least a k\Gamma1 k (1 + 1 2\Delta+k\Gamma1 ) fraction of the edges, improving previous bounds known. The partitions also apply to constraint networks, for which we give a tight analysis of natural local search heuristics for the maximum constraint satisfaction problem. These partitions also imply efficient approximations for several problems on weighted bounded-degree graphs. In particular, we improve the best performance ratio for the weighted independent set problem to 3 \Delta+2 , and obtain an efficient algorithm for coloring 3-colorable graphs with at most 3\Delta+2 4 colors

Vertex colouring and forbidden subgraphs - a survey

There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions

Determination of a Graph\u27s Chromatic Number for Part Consolidation in Axiomatic Design

Mechanical engineering design practices are increasingly moving towards a framework called axiomatic design (AD). A key tenet of AD is to decrease the information content of a design in order to increase the chance of manufacturing success. An important way to decrease information content is to fulfill multiple functional requirements (FRs) by a single part: a process known as part consolidation. One possible method for determining the minimum number of required parts is to represent a design by a graph, where the vertices are the FRs and the edges represent the need to separate their endpoint FRs into separate parts. The answer is then the chromatic number of such a graph. This research investigates the suitability of using two existing algorithms and a new algorithm for finding the chromatic number of a graph in a part consolidation tool that can be used by designers. The runtime complexities and durations of the algorithms are compared empirically using the results from a random graph analysis with binomial edge probability. It was found that even though the algorithms are quite different, they all execute in the same amount of time and are suitable for use in the desired design tool