Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Replication in critical graphs and the persistence of monomial ideals   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.

27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
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The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs