Properties of this class of graphs I am interested in a certain class of graphs but have very little graph theory background, I was hoping that you guys could poke me in the right direction. The class of graphs is as follows:


*

*Multi-graph

*Connected components are directed Euler graphs

*Minimal degree is $4$

*Bipartite

*We are given an embedding in $X$, a compact orientable surface of genus $g$. 

*The faces of the graph (when considering said embedding) are $4$-colourable.


I'm interested in any property of these graphs!
Thanks in advance!
 A: OP has clarified in the comments that some equations are wanted. 
Assuming that the embedding in $X$ is a 2-cell embedding (that is, that each region is homeomorphic to a disk) we have Euler's formula, $$v-e+f=2-2g$$ where $v,e,f$ are the number of vertices, edges, and faces, respectively.  
For any graph, $2e=\sum_x d(x)$, the sum over all vertices $x$ of the degree of $x$. Since the minimal degree is 4, we get $$e\ge2v$$
I think the 4-colorability gives another inequality relating these invariants, but I'm not remembering how to find it.   
A: This is not a complete answer, but a sketchy response to one of Gerry Myerson's comments.  Yes, infinitely many graphs satisfy the listed conditions.
Fisk and Mohar [JCTA 1994] proved that any undirected graph $G$ with the following properties is 4-colorable:


*

*The shortest cycle in $G$ has length at least four.

*$G$ can be embedded on a surface of genus $g$.

*The shortest separating cycle in the embedding of $G$ has length at least $1000\,\log_2(g+1)$.  A cycle is separating if it cuts the surface into two pieces.  (The constant $1000$ is a conservative estimate; Fisk and Mohar prove the existence of an appropriate constant, but don't give any explicit value.)


It is fairly easy to construct arbitrarily large graphs $G$ that satisfy these requirements by subdividing any triangulation of the surface.  The faces of the dual graph $G^*$ are 4-colorable (by definition of "dual").  Subdividing each edge of $G^*$ with a new vertex makes the dual graph bipartite.  Finally, replacing each undirected edge with a pair of opposing directed edges makes the graph Eulerian with minimum degree $4$.
