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Article Dans Une Revue Journal of Combinatorial Theory, Series A Année : 2014

Replication in critical graphs and the persistence of monomial ideals

Résumé

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.
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Dates et versions

hal-00946588 , version 1 (17-02-2014)

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Tomas Kaiser, Matej Stehlik, Riste Škrekovski. Replication in critical graphs and the persistence of monomial ideals. Journal of Combinatorial Theory, Series A, 2014, 123 (1), pp.239-251. ⟨10.1016/j.jcta.2013.12.005⟩. ⟨hal-00946588⟩
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