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arXiv · 1301.6983

Replication in critical graphs and the persistence of monomial ideals

Tomáš Kaiser·Matěj Stehlík·Riste Škrekovski
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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.

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Tomáš Kaiser, Matěj Stehlík, Riste Škrekovski. 2013-01-29. Replication in critical graphs and the persistence of monomial ideals. https://doi.org/10.1016/j.jcta.2013.12.005

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