Elsevier

Discrete Mathematics

Volume 144, Issues 1–3, 8 September 1995, Pages 23-31
Discrete Mathematics

Colorings of diagrams of interval orders and α-sequences of sets

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Abstract

We show that a proper coloring of the diagram of an interval order I may require 1 + ⌈log2 height(I)⌉ colors and that 2 + ⌈log2 height(I)⌉ colors always suffice. For the proof of the upper bound we use the following fact: A sequence C1, …, Ch of sets (of colors) with the property Cj⊈Ci-1∪Ci for all 1<i<j⩽h can be used to color the diagram of an interval order with the colors of the Ci. We construct α-sequences of length 2n − 2 + ⌊(n − 1)/2)⌋ using n colors. The length of α-sequences is bounded by 2n − 1 + ⌊(n − 1)/2)⌋ and sequences of this length have some nice properties. Finally we use α-sequences for the construction of long cycles between two consecutive levels of the Boolean lattice. The best construction known until now could guarantee cycles of length Ω(Nc) where N is the number of vertices and c ≈ 0.85. We exhibit cycles of length 14 N.

Keywords

Interval order
Diagram
Chromatic number
Hamiltonian path
Boolean lattice

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Partially supported by the DFG.