Homeomorphically irreducible spanning trees in hexangulations of surfaces

A homeomorphically irreducible spanning tree (HIST) of a connected graph is a spanning tree without vertices of degree two. The determination of the existence problem of a homeomorphically irreducible spanning tree in a plane cubic graph is NP-complete. A hexangulation of a surface is a cubic graph embedded on a surface such that every face is bounded by a hexagon. It is a problem asked by Hoffmann-Ostenhof and Ozeki that whether there are finitely or infinitely many hexangulations of torus with homeomorphically irreducible spanning trees. In this paper, we show that a family of hexangulations of surfaces, denoted by H(m,n), have a homeomorphically irreducible spanning tree if and only if it has an odd number of faces, which answers the problem of Hoffmann-Ostenhof and Ozeki for hexangulations of surfaces.