Folding

We define folding of a directed graph as a coloring (or a homomorphism) which is injective on all the down sets of a given depth. While in general foldings are as complicated as homomorphisms for some classes they present useful tool to study colorings and homomorphisms. Our main result yields for any proper minor closed class K a folding (of any prescribed depth) using a fixed number of colors. This in turn yields (for any K) the existence of a Kk-free graph which bounds (in the homomorphism order) all Kk-free graphs belonging to K. This has been conjectured in [J. Nešetřil, Aspects of structural combinatorics, Taiwanese J. Math. 3 (4) (1999) 381–424] and elsewhere and solved for k=3 in [J. Nešetřil, P. Ossona de Mendez, Colorings and homomorphisms of minor closed classes, in: J. Pach, et al. (Eds.), J. Goodman and R. Pollack Festschrift, Springer, Berlin, 2003, pp. 651–664]. Particularly, we prove (without using 4CT) the existence of a graph H which satisfies χ(H)⩽5, ω(H)⩽4 and such that any planar graph G is homomorphic to H. This is sandwiched between 4CT and 5CT for planar graphs and the general case has bearing to Hadwiger conjecture.