Proper connection with many colors

In this work, we consider only edge-colorings of graphs. Since Vizing’s fundamental result [9], proper edge colorings of graphs, colorings such that no two adjacent edges have the same color, have become an essential topic for every beginning graph theorist. Proper edge colorings have many applications in signal transmission [7], bandwidth allocation [3] and many other areas [5, 6, 8]. See [1] for a survey of the case where two colors (where the term ‘alternating’ can be used in place of ‘proper’) are used. Since many of these applications depend only on properly colored substructures of the graph, not necessarily properly coloring the entire graph, it is natural to restrict our attention to subgraphs. If a graph is properly colored, then every subgraph is properly colored. We relax this condition by requiring only some of the subgraphs to be properly colored. In particular, we say that a colored graph is properly connected if, between every pair of vertices, there exists a properly colored path. As defined in [2], the proper connection number of a graph pc(G) is the minimum number of colors k such that there exists a k coloring of G which is properly connected. As a specific application of proper connectivity to network security, suppose a network administrator would like to create a more secure network.