Replication in critical graphs and the persistence of monomial ideals

  • Authors:

  • Tomáš Kaiser;Matj Stehlík;Riste Škrekovski

  • Affiliations:

  • Department of Mathematics, Institute for Theoretical Computer Science (CE-ITI) and European Centre of Excellence NTIS-New Technologies for Information Society, University of West Bohemia, Univerzi ...;UJF-Grenoble 1/CNRS/Grenoble-INP, G-SCOP UMR5272, Grenoble, F-38031, France;Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia and Faculty of Information Studies, Novo Mesto, Slovenia and FAMNIT, University of Primorska, Koper, Slovenia

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2014

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Abstract

Motivated by questions about square-free monomial ideals in polynomial rings, in 2010 Francisco et al. conjectured that for every positive integer k and every k-critical (i.e., critically k-chromatic) graph, there is a set of vertices whose replication produces a (k+1)-critical graph. (The replication of a set W of vertices of a graph is the operation that adds a copy of each vertex w in W, one at a time, and connects it to w and all its neighbours.) We disprove the conjecture by providing an infinite family of counterexamples. Furthermore, the smallest member of the family answers a question of Herzog and Hibi concerning the depth functions of square-free monomial ideals in polynomial rings, and a related question on the persistence property of such ideals.