Pisanski's conjecture concerns graph embeddings on orientable surfaces of regular
graphs that are bipartite, with quadrilateral
faces. For integers 
, it asserts that a simple connected
graph with 
vertices in each part and vertex
degree 
at every vertex can be chosen with a cellular orientable
graph embedding whose face boundaries are simple
cycle graphs of length four iff 
(Ulrigg and Metzger 2026).
Ulrigg and Metzger (2026) proved the conjecture by explicit constructions. For 
, their graph 
has minimum orientable
genus
 |
The divisibility condition is necessary by the Euler characteristic formula. The orientable quadrangulation attains the lower bound on orientable genus for a simple bipartite graph with these vertex degrees.
