Irreducible quadrangulations of the torus (Q1924143)

A quadrangulation \(G\) of a closed surface \(S\) is a simple graph embedded in \(S\) all whose faces are quadrangles. A contraction at a pair of points \(\{b,d\}\) which lie in a common face \(f\) is a combinatorial operation that essentially removes the quadrangle \(f\) and ``glues'' together faces containing \(b\) or \(d\) (see precise definition in the paper). A quadrangulation is irreducible if no further contraction at a pair of points is possible. This paper finds all the irreducible triangulations of the torus (continuing previous work about the sphere and projective plane). There are 8 possible cases and this classification implies that any two quadrangulations of the torus with same number of vertices that are either both bipartite or both non-bipartite can be transformed into one another using diagonal slides and diagonal rotations (these are very easy operations on the edges). Another application is to find the 7 minor-minimal 2-representative graphs on the torus.