The Euler genus of a closed connected surface 
is 
,
where 
is its Euler
characteristic. For an orientable surface
of genus 
this is 
, while for a nonorientable
surface it is the number of cross-caps. Thus the
sphere has Euler genus 0, the real
projective plane has Euler genus 1, and the torus and
Klein bottle have Euler genus 2.
The Euler genus of a graph is the minimum Euler genus of a surface in which it embeds.
This is the smaller of twice its orientable genus 
and its nonorientable
genus 
,
namely 
.
It can therefore differ from twice the graph genus
because an embedding in a nonorientable
surface is allowed (Mohar 1998). Graph basis
number admits bounds in terms of Euler genus (Knauer 2026).
