European Journal of Combinatorics
Good and semi-strong colorings of oriented planar graphs
Information Processing Letters
Gro¨tzsch's 3-color theorem and its counterparts for the torus and the projective plane
Journal of Combinatorial Theory Series B
Homomorphisms of Edge-Colored Graphs and Coxeter Groups
Journal of Algebraic Combinatorics: An International Journal
Colored homomorphisms of colored mixed graphs
Journal of Combinatorial Theory Series B
Universal H-colorable graphs without a given configuration
Discrete Mathematics
WG '01 Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science
Homomorphism bounded classes of graphs
European Journal of Combinatorics
K5-free bound for the class of planar graphs
European Journal of Combinatorics - Special issue on Eurocomb'03 - graphs and combinatorial structures
Grad and classes with bounded expansion III. Restricted graph homomorphism dualities
European Journal of Combinatorics
Finite dualities and map-critical graphs on a fixed surface
Journal of Combinatorial Theory Series B
Survey: Colouring, constraint satisfaction, and complexity
Computer Science Review
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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 κ a folding (of any prescribed depth) using a fixed number of colors. This in turn yields (for any κ) the existence of a Kk-free graph which bounds (in the homomorphism order) all Kk-free graphs belonging to κ. 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 beating to Hadwiger conjecture.