Chromatic Number of the Odd-Distance Graph

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Chromatic Number of the Odd-Distance Graph

• Jacob Steinhardt

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Odd-Distance Graph

• Make a graph where the vertices consist of the
  points in the plane, and two points are connected
  if their distance is an odd-integer, i.e. if
  sqrt((x_1-x_2)2(y_1-y_2)2) 2k1 for some k.
• Question Is the chromatic number of this graph
  finite?

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Chromatic Number

• The chromatic number is the minimum number of
  colors we need to color the vertices of a graph
  so that no two adjacent vertices have the same
  color.
• E.g. The four-color theorem asserts that the
  chromatic number of all planar graphs is at most
  four.
• The problem is equivalent to asking if we can
  color the points in the plane with finitely many
  colors so that no two points of the same color
  are at an odd integral distance from each other.

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Ideas

• Let A be the adjacency matrix of a graph, and let
  lambda_max and lambda_min be the largest and
  smallest eigenvalues. Then
• Chi gt 1-lambda_max/lambda_min
• Chi is chromatic number
• For infinite graphs, only applies to measurable
  colorings
• Idea Find a sequence of weightings for our graph
  such that lambda_max/lambda_min -gt -infinity

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Weighting

• If we weight vertices exponentially by their
  distance (i.e. as alpha-dist), then we can prove
  that lambda_max/lambda_min -gt -infinity as alpha
  -gt 1
• This shows that no finite measurable coloring
  exists

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Non-measurable colorings

• If we can find a sequence of finite subgraphs
  whose adjacency matrices in some sense converge
  to that for our infinite graph, we may be able to
  get our result to hold for non-measurable
  colorings as well, as all colorings on a finite
  graph are measurable.
• Unsure how to do this...maybe use the coherent
  topology on our graph, but it doesnt have all
  the properties one might hope for (and I also
  dont know enough topology)?