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Journal of Combinatorial Theory Series B
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Journal of Combinatorial Theory Series B
Article . 2006
License: Elsevier Non-Commercial
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Journal of Combinatorial Theory Series B
Article . 2006 . Peer-reviewed
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Folding

Authors: Jaroslav Nesetril; Patrice Ossona de Mendez;
Abstract

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.

Keywords

homomorphism, Graph minors, constrained orientation, Theoretical Computer Science, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], Coloring of graphs and hypergraphs, bound, minor, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, coloring

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
12
Average
Top 10%
Top 10%
hybrid