Folding

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 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 some classes they present an useful tool to study colorings and homomorphisms. Our main result yields for any proper minor closed class C a folding (of any prescribed depth) using a fixed number of colors. This in turn yields (for any C) the existence of a Kk-free graph which bounds all Kk-free graphs belonging to C. This has been conjectured and solved for k=3. Particularly, we prove (without using 4CT) the existence of a graph H with chromatic number at most 5 and clique number at most 4, 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.

Mots clés: constrained orientation; homomorphism

Domaines: Combinatoire [math.CO]

Jaroslav Nesetril, Patrice Ossona de Mendez. Folding. Journal of Combinatorial Theory, Series B, 2006, 96-5, pp.730-739. ⟨10.1016/j.jctb.2006.02.002⟩. ⟨hal-00081706⟩