2013/12/22 by Friesen, Tyler, Manturov, Vassily Olegovich
#57C27 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Primary 05C10 #Secondary 57C15
paper · doi:10.48550/arxiv.1312.6342
This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper by the authors, the problem of calculating whether a given *-graph in which all vertices have degree 4 or 6 admits a Z2-homologically trivial embedding into a given orientable surface was shown to be equivalent to a problem on matrices. Here we extend those results to nonorientable surfaces. The embeddability condition that we obtain yields quadratic-time algorithms to determine whether a *-graph with all vertices of degree 4 or 6 admits a Z2-homologically trivial embedding into the projective plane or into the Klein bottle.