Convex excess in partial cubes

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Authors

Colleges, School and Institutes

Abstract

The convex excess ce(G) of a graph G is introduced as inline image 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 inline image 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 inline image r et al. [Tiled partial cubes, J Graph Theory 40 (2002) 91–103] does not hold. Copyright © 2011 John Wiley & Sons, Ltd.

Details

Original languageEnglish
Pages (from-to)356-369
Number of pages14
JournalJournal of Graph Theory
Volume69
Issue number4
Early online date3 Mar 2011
Publication statusPublished - 1 Apr 2012