Tree depth, subgraph coloring and homomorphism bounds

Jaroslav Nesetril (1) , Patrice Ossona de Mendez (2)

1 KAM - Department of Applied Mathematics (KAM)
2 CAMS - Centre d'Analyse et de Mathématique sociales

We define the notions tree depth and upper chromatic number of a graph and show their relevance to local - global problems for graphs partitions. Particularly 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. Particularly, we prove the following result: For every graph M and a finite set F of connected graphs there exists a (universal) graph U = U(M, F) ∈ Forb(F) such that any graph G ∈ Forb(F) which does not have M as a minor satisfies G → U (i.e. is homomorphic to U). 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.

Mots clés: tree depth; rank; coloring; homomorphism; minor closed classes; constrained orientation

Domaines: Combinatoire [math.CO]

Jaroslav Nesetril, Patrice Ossona de Mendez. Tree depth, subgraph coloring and homomorphism bounds. European Journal of Combinatorics, 2006, 27(6), pp.1022-1041. ⟨10.1016/j.ejc.2005.01.010⟩. ⟨hal-00023821⟩