Embedding grids in surfaces

Authors:
J. F. Geelen;R. B. Richter;G. Salazar
Affiliations:
Department of Combinatorics & Optimization, University of Waterloo, Waterloo, ON, Canada N2L 3G1;Department of Combinatorics & Optimization, University of Waterloo, Waterloo, ON, Canada N2L 3G1;Instituto de Investigaciones en Comunicaciones Ópticas, Universidad Autonóma de San Luis Potosí San Luis Potosí SLP Mexico, 78210 Mexico
Venue:
European Journal of Combinatorics - Special issue: Topological graph theory
Year:
2004

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Abstract

We show that if a very large grid is embedded in a surface, then a large subgrid is embedded in a disc in the surface. This readily implies that: (a) a minor-minimal graph that does not embed in a given surface has no very large grid; and (b) a minor-minimal k-representative embedding in the surface has no very large grid. Similar arguments show (c) that if G is minimal with respect to crossing number, then G has no very large grid. This work is a refinement of Thomassen (J. Combin. Theory Ser. B 70 (1997) 306).