The Order Upper Bound on Parity Embedding of a Graph

A graphΓisparity embeddedin a surface if a closed path in the graph is orientation preserving or reversing according to whether its length is even or odd. Theparity demigenusofΓis the minimum of 2−χ(S) (whereχis the Euler characteristic) over all surfacesSin whichΓcan be parity embedded. We calculate the maximum parity demigenus over all graphs, simple or not, of ordern.