Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bounds for identifying codes in terms of degree parameters   An identifying code is a subset of vertices of a graph such that each vertex
is uniquely determined by its neighbourhood within the identifying code. If
$\M(G)$ denotes the minimum size of an identifying code of a graph $G$, it was
conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there
exists a constant $c$ such that if a connected graph $G$ with $n$ vertices and
maximum degree $d$ admits an identifying code, then $\M(G)\leq
n-\tfrac{n}{d}+c$. We use probabilistic tools to show that for any $d\geq 3$,
$\M(G)\leq n-\tfrac{n}{\Theta(d)}$ holds for a large class of graphs
containing, among others, all regular graphs and all graphs of bounded clique
number. This settles the conjecture (up to constants) for these classes of
graphs. In the general case, we prove $\M(G)\leq n-\tfrac{n}{\Theta(d^{3})}$.
In a second part, we prove that in any graph $G$ of minimum degree $\delta$ and
girth at least 5, $\M(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n$.
Using the former result, we give sharp estimates for the size of the minimum
identifying code of random $d$-regular graphs, which is about $\tfrac{\log
d}{d}n$.

29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat