The Order Upper Bound on Parity Embedding of a Graph

Let us try to embed a graph 1, not necessarily simple, in a surface so that every odd polygon (the graph of a simple closed path of odd length), regarded as a path in the surface, reverses orientation while every even polygon preserves it. What is the smallest surface in which this is possible? That is, what is the minimum demigenus d(S)=2&/(S) over all embedding surfaces S? We call this kind of embedding parity embedding and the smallest d(S) the parity demigenus of 1, written d(&1 ). There is in general no exact formula but there is a simple lower bound based on Euler's polyhedral formula and the obvious fact that a face boundary must (with trivial exceptions) have length at least 4: