Finite dualities and map-critical graphs on a fixed surface

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Let K be a class of graphs. A pair ( F , U ) is a finite duality in K if U ∈ K , F is a finite set of graphs, and for any graph G in K we have G ⩽ U if and only if F ⩽̸ G for all F ∈ F , where “⩽” is the homomorphism order. We also say U is a dual graph in K . We prove that the class of planar graphs has no finite dualities except for two trivial cases. We also prove that the class of toroidal graphs has no 5-colorable dual graphs except for two trivial cases. In a sharp contrast, for a higher genus orientable surface S we show that Thomassenʼs result (Thomassen, 1997 [17]) implies that the class, G ( S ) , of all graphs embeddable in S has a number of finite dualities. Equivalently, our first result shows that for every planar core graph H except K 1 and K 4 , there are infinitely many minimal planar obstructions for H-coloring (Hell and Nešetřil, 1990 [4]), whereas our later result gives a converse of Thomassenʼs theorem (Thomassen, 1997 [17]) for 5-colorable graphs on the torus.