Euler Cycles inK2mPlus Perfect Matching

We1999 Academic Pressanalyze the freedom one has when constructing an Euler cycle throughCopyright K2m+, the complete graph of even order plus a perfect matching. We start with the fact that a multigraphGwithmedges is Eulerian iff it can be obtained from a cycle of lengthmby glueing certain vertices together. Our extension is the following. We deal with cycles of (2m2) + medges, whose vertices are colored black and white. There are 2mblack vertices and (2m2) − mwhite vertices. Assuming that glueing a black and a white vertex results in a black vertex we describe a method of how to glue certain vertices of a given cycle in such a way that the resulting multigraph isK2m+with all vertices black.