Chromatic number of the n-cube

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Discussion Overview

The discussion centers around the chromatic number of the n-cube, specifically in the context of graph theory. Participants explore the properties of the n-cube as a bipartite graph and its implications for coloring.

Discussion Character

  • Technical explanation, Conceptual clarification

Main Points Raised

  • One participant asks about the chromatic number of the n-cube, noting that for lower dimensions like the square and 3-cube, it is 2.
  • Another participant mentions empirical evidence suggesting that the chromatic number is consistently 2.
  • A third participant asserts that the n-cube is bipartite, which leads to a chromatic number of 2, and provides a method for partitioning the vertices based on the parity of the number of 1's in their binary representation.
  • A later reply expresses gratitude for the clarification provided.

Areas of Agreement / Disagreement

Participants appear to agree that the chromatic number of the n-cube is 2, primarily based on its bipartite nature. However, the discussion does not explore any competing views or alternative interpretations.

Dragonfall
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What is the chromatic number of the n-cube? As a graph, I mean. For the square and 3-cube, for example, it's 2.
 
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Empirical evidence suggests it's just 2.
 
The n-cube is bipartite, so its chromatic number is 2. If we label the vertices canonically with vectors in [tex]\{0, 1\}^n[/tex], then we can partition the vertices into those with an even number of 1's and those with an odd number of 1's.
 
Oh ya, thanks.
 

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