Title:  Edge Colored Graphs: Structural Results, Algorithms and Applications

Recent years have seen significant interests in problems of colored graphs due to their theoretical appeal and applicability in various fields. In particular, problems arising in molecular biology are often formulated using colored graphs, i.e. graphs with colored edges and/or vertices. Given such a graph, original problems correspond to extracting sub-graphs such as Hamiltonian and Eulerian paths or cycles colored in a specified pattern. The most natural pattern in such a context is the so-called proper coloring, i.e. adjacent edges/vertices having different colors. Properly colored paths and cycles have applications in various other fields such as VLSI for compacting a programmable logical array. Other applications include social sciences where a color represents a relation between two individuals and the notion of properly edge colored paths and cycles is related to the balance of a graph. Although a large body of work has already been done, in much of the research, the number of colors is restricted to two. For instance, while how to find efficiently a properly edge colored Hamiltonian cycle in a 2-edge colored complete graph is well known, it is a long standing open question how to find such cycles in complete graphs whose edges are colored by any number of colours. During this talk, the most important results in the area related to various forms of proper subgraphs (trees, paths, cycles etc) in complete or general graphs for any number of used colors will be discussed.