Convex Excess in Partial Cubes

  • Authors:

  • Sandi Klavžar;Sergey Shpectorov

  • Affiliations:

  • Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia and Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160 2000 ...;School of Mathematics, University of Birmingham Edgbaston, Birmingham B15 2TT, United Kingdom

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2012

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Abstract

The convex excess ce(G) of a graph G is introduced as where the summation goes over all convex cycles of G. It is proved that for a partial cube G with n vertices, m edges, and isometric dimension i(G), inequality 2n−m−i(G)−ce(G)≤2 holds. Moreover, the equality holds if and only if the so-called zone graphs of G are trees. This answers the question from Bre r et al. [Tiled partial cubes, J Graph Theory 40 (2002) 91–103] whether partial cubes admit this kind of inequalities. It is also shown that a suggested inequality from Bre r et al. [Tiled partial cubes, J Graph Theory 40 (2002) 91–103] does not hold. Copyright © 2011 John Wiley & Sons, Ltd. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: Ministry of Science of Slovenia; Contract grant number: P1-0297 (to S. K.); Contract grant sponsor: NSA (S. S.).)