The exact isoperimetric inequality for ternary and quaternary cubes

  • Authors:

  • Tomaz Slivnik

  • Affiliations:

  • Department of Pure Mathematics and Mathematical Statistics, Cambridge University, 16 Mill Lane, Cambridge CB2 1SB, UK

  • Venue:
  • Discrete Mathematics - Algebraic and topological methods in graph theory
  • Year:
  • 2002

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Abstract

We extend the well-known edge-isoperimetric inequality of Harper, Bernstein and Hart to ternary and quaternary cubes. More generally, let Q be the graph with vertex set V = Πi=1n [Ki] in which x ∈ V is joined to y ∈ V if for some i we have |xi - yi| = 1 and xj = yj for all j ≠ i. If k1 ≥ ... ≥ kn and k2 ≤ 4, we prove that for any 0 ≤ m ≤ |V|, no m-set of vertices of Q is joined to the rest of Q by fewer edges than the set of the first m vertices of Q in the lexicographic ordering on V.