The concept of holonomy comes from differential geometry. It describes the behaviour of a vector on a surface when it is moved via parallel transport along a closed curve on the surface.
A similar phenomenon can be observed on a planar graph whose faces are all quadrilaterals. Although this context is free of geometric notions such as "vector," "orientation," and "parallel transport", we can come up with some combinatorial analogs. I won't define these precisely, but hope the pictures below give the main idea. In fact, my question is: is there literature which has formalized and proven the following observation?
Three quadrilateral planar graphs are shown. In each, there is an initial blue "vector". This vector is transformed via "parallel transport" through a (counterclockwise) cycle of boundary faces, resulting in a final red vector in the initial face. This final vector has one of four orientations. This orientation, in terms of "quarter turns" from the initial vector corresponds with the total degree of the interior vertices, modulo 4.
It the triangular graph, for example, there is one interior vertex of degree 3, which is 3 (or -1) modulo 4. And the final vector differs from the initial vector by 3 (or -1) quarter turns.
In the second graph all interior vertices have degree 4, so the final vector has the same orientation as the initial. (Although you could argue that it has undergone 4 quarter turns, and that the number of turns should be taken modulo 4.)
In the third graph there are two interior vertices of degree 3, and the final vector differs from the initial by two quarter turns.
Wikipedia refers to a combinatorial analog of the Gauss Bonnet theorem and I was wondering whether somewhere in the literature this combinatorial analog of holonomy has been worked out as just another version of Stokes Theorem or the like.