Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nathanson's Heights and the CSS Conjecture for Cayley Graphs   Let $G$ be a finite directed graph, $\beta(G)$ the minimum size of a subset
$X$ of edges such that the graph $G' = (V,E \smallsetminus X)$ is directed
acyclic and $\gamma(G)$ the number of pairs of nonadjacent vertices in the
undirected graph obtained from $G$ by replacing each directed edge with an
undirected edge. Chudnovsky, Seymour and Sullivan \cite{CSS07} proved that if
$G$ is triangle-free, then $\beta(G) \leq \gamma(G)$. They conjectured a
sharper bound (so called the "CSS conjecture") that $\beta(G) \leq
\dfrac{\gamma(G)}{2}$. Nathanson and Sullivan verified this conjecture for the
directed Cayley graph $\Cay(\bbZ/N\bbZ, E_A)$ whose vertex set is the additive
group $\bbZ/N\bbZ$ and whose edge set $E_A$ is determined by $E_A = {(x,x+a) :
x \in \bbZ/N\bbZ, a \in A}$ when $N$ is prime in \cite{NS07} by introducing
"height". In this work, we extend the definition of height and the proof of CSS
conjecture for $\Cay(\bbZ/N\bbZ, E_A)$ to any positive integer $N$.

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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat