A theorem on graph embedding with a relation to hyperbolic volume

We prove that a planar cubic cyclically 4-connected graph of odd χ < 0 is the dual of a 1-vertex triangulation of a closed orientable surface. We explain how this result is related to (and applied to prove at a separate place) a theorem about hyperbolic volume of links: the maximal volume of alternating links of given χ < 0 does not depend on the number of their components.