Finite dualities and map-critical graphs on a fixed surface

Let K be a class of graphs. Then, K is said to have a finite duality if there exists a pair (F , U), where U ∈ K and F is a finite set of graphs, such that for any graph G in K we have G ≤ U if and only if F 6≤ G for all F ∈ F (“ ≤ ” is the homomorphism order). We prove that the class of planar graphs has no finite duality except for two trivial cases. We also prove that a 5-colorable toroidal graph U obtains a finite duality on a given fixed surface if and only if the core of U is K5. In a sharp contrast, for a higher genus orientble surface S we show that Thomassen’s result [15] 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, whereas our later result gives a converse of Thomassen’s theorem [15] for 5-colorable graphs on the torus. Supported by a Grant 1M002160808 of Czech Ministry of Education and AEOLUS Partially supported by DIMATIA and by a Grant 1M002160808