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Harmonious and achromatic colorings of fragmentable hypergraphs.
- Source :
-
European Journal of Combinatorics . Dec2017, Vol. 66, p60-80. 21p. - Publication Year :
- 2017
-
Abstract
- A harmonious coloring of a k -uniform hypergraph H is a rainbow vertex coloring such that each k -set of colors appears on at most one edge. A rainbow coloring of H is achromatic if each k -set of colors appears on at least one edge. The harmonious number (resp. achromatic number ) of H , denoted by h ( H ) (resp. ψ ( H ) ) is the minimum (resp. maximum) possible number of colors in a harmonious (resp. achromatic) coloring of H . A class H of hypergraphs is fragmentable if for every H ∈ H , H can be fragmented into components of a bounded size by removing a “small” fraction of vertices. We show that for every fragmentable class H of bounded degree hypergraphs, for every ϵ > 0 and for every hypergraph H ∈ H with m ≥ m 0 ( H , ϵ ) edges we have h ( H ) ≤ ( 1 + ϵ ) k ! m k and ψ ( H ) ≥ ( 1 − ϵ ) k ! m k . As corollaries, we answer a question posed by Blackburn concerning the maximum length of t -subset packing sequences of constant radius and derive an asymptotically tight bound on the minimum number of colors in a vertex-distinguishing edge coloring of cubic planar graphs, which is a step towards confirming a conjecture of Burris and Schelp. [ABSTRACT FROM AUTHOR]
- Subjects :
- *HYPERGRAPHS
*COLORING matter
*GRAPH theory
*GRAPHIC methods
*PLANAR graphs
Subjects
Details
- Language :
- English
- ISSN :
- 01956698
- Volume :
- 66
- Database :
- Academic Search Index
- Journal :
- European Journal of Combinatorics
- Publication Type :
- Academic Journal
- Accession number :
- 124936522
- Full Text :
- https://doi.org/10.1016/j.ejc.2017.06.013