Given a graph G, its k-coloring graph is the graph whose vertex set is the proper k-colorings of the vertices of G with two k−colorings adjacent if they differ at exactly one vertex. In this paper, we consider the question: Which graphs can be coloring graphs? In other words, given a graph H, do there exist G and k such that H is the k-coloring graph of G? We will answer this question for several classes of graphs and discuss important obstructions to being a coloring graph involving order, girth, and induced subgraphs.
Through eigenanalysis of communication matrices, we develop a new objective function formulation for mapping tasks to parallel computers with cellular networks. This new formulation significantly speeds up the solution process through consideration of the symmetries in the supply matrix of a network and a transformation of the demand matrix of any application. The extent of the speedup is not easily obtainable through analytical means for most production networks. However, numerical experiments of mapping wave equations on 2D mesh onto 3D torus networks by simulated annealing demonstrate a far superior convergence rate and quicker escape from local minima with our new formulation than with the standard graph theory-based one.
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