Finite Dualities and Map-Critical Graphs on a Fixed Surface

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 Thomassen\u27s 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 K1 and K4, there are infinitely many minimal planar obstructions for H-coloring (Hell and Nešetřil, 1990 [4]), whereas our later result gives a converse of Thomassen\u27s theorem (Thomassen, 1997 [17]) for 5-colorable graphs on the torus