Abstract
An edge-colored graph G is properly colored if no two adjacent edges share a color in G. An edge-colored connected graph G is properly connected if between every pair of distinct vertices, there exists a path that is properly colored. In this paper, we discuss how to make a connected graph properly connected efficiently. More precisely, we consider the problem to convert a given monochromatic graph into properly connected by recoloring p edges with q colors so that $$p+q$$ is as small as possible. We discuss how this can be done efficiently for some restricted graphs, such as trees, complete bipartite graphs and graphs with independence number 2.
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