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Properly Coloured Cycles and Paths: Results and Open Problems (0805.3901v3)
Published 26 May 2008 in cs.DM and cs.DS
Abstract: In this paper, we consider a number of results and seven conjectures on properly edge-coloured (PC) paths and cycles in edge-coloured multigraphs. We overview some known results and prove new ones. In particular, we consider a family of transformations of an edge-coloured multigraph $G$ into an ordinary graph that allow us to check the existence PC cycles and PC $(s,t)$-paths in $G$ and, if they exist, to find shortest ones among them. We raise a problem of finding the optimal transformation and consider a possible solution to the problem.