Tree-depth, subgraph coloring and homomorphism bounds

We define the notions tree-depth and upper chromatic number of a graph and show their relevance to local–global problems for graph partitions. In particular we show that the upper chromatic number coincides with the maximal function which can be locally demanded in a bounded coloring of any proper minor closed class of graphs. The rich interplay of these notions is applied to a solution of bounds of proper minor closed classes satisfying local conditions. In particular, we prove the following result:For every graph MM and a finite set FF of connected graphs there exists a (universal) graph U=U(M,F)∈Forbh(F) such that any graph G∈Forbh(F) which does not have MM as a minor satisfies G⟶UG⟶U (i.e. is homomorphic to UU).This solves the main open problem of restricted dualities for minor closed classes and as an application it yields the bounded chromatic number of exact odd powers of any graph in an arbitrary proper minor closed class. We also generalize the decomposition theorem of DeVos et al. [M. DeVos, G. Ding, B. Oporowski, D.P. Sanders, B. Reed, P. Seymour, D. Vertigan, Excluding any graph as a minor allows a low tree-width 2-coloring, J. Combin. Theory Ser. B 91 (2004) 25–41].