Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Minimizing the numbers of cliques and cycles of fixed size in an
  $F$-saturated graph   This paper considers two important questions in the well-studied theory of
graphs that are $F$-saturated. A graph $G$ is called $F$-saturated if $G$ does
not contain a subgraph isomorphic to $F$, but the addition of any edge creates
a copy of $F$. We first resolve a fundamental question of minimizing the number
of cliques of size $r$ in a $K_s$-saturated graph for all sufficiently large
numbers of vertices, confirming a conjecture of Kritschgau, Methuku, Tait, and
Timmons. We also go further and prove a corresponding stability result. Next we
minimize the number of cycles of length $r$ in a $K_s$-saturated graph for all
sufficiently large numbers of vertices, and classify the extremal graphs for
most values of $r$, answering another question of Kritschgau, Methuku, Tait,
and Timmons for most $r$. We then move on to a central and longstanding
conjecture in graph saturation made by Tuza, which states that for every graph
$F$, the limit $\lim_{n \rightarrow \infty} \frac{\sat(n, F)}{n}$ exists, where
$\sat(n, F)$ denotes the minimum number of edges in an $n$-vertex $F$-saturated
graph. Pikhurko made progress in the negative direction by considering families
of graphs instead of a single graph, and proved that there exists a graph
family $\mathcal{F}$ of size $4$ for which $\lim_{n \rightarrow \infty}
\frac{\sat(n, \mathcal{F})}{n}$ does not exist (for a family of graphs
$\mathcal{F}$, a graph $G$ is called $\mathcal{F}$-saturated if $G$ does not
contain a copy of any graph in $\mathcal{F}$, but the addition of any edge
creates a copy of a graph in $\mathcal{F}$, and $\sat(n, \mathcal{F})$ is
defined similarly). We make the first improvement in 15 years by showing that
there exist infinitely many graph families of size $3$ where this limit does
not exist. Our construction also extends to the generalized saturation problem
when we minimize the number of fixed-size cliques.

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The categories of this paper are: bayes-an
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
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The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
9
The categories of this paper are: nlin
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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: q-alg
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph