Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Proper disconnection of graphs   For an edge-colored graph $G$, a set $F$ of edges of $G$ is called a
\emph{proper cut} if $F$ is an edge-cut of $G$ and any pair of adjacent edges
in $F$ are assigned by different colors. An edge-colored graph is \emph{proper
disconnected} if for each pair of distinct vertices of $G$ there exists a
proper edge-cut separating them. For a connected graph $G$, the \emph{proper
disconnection number} of $G$, denoted by $pd(G)$, is the minimum number of
colors that are needed in order to make $G$ proper disconnected. In this paper,
we first give the exact values of the proper disconnection numbers for some
special families of graphs. Next, we obtain a sharp upper bound of $pd(G)$ for
a connected graph $G$ of order $n$, i.e, $pd(G)\leq \min\{ \chi'(G)-1, \left
\lceil \frac{n}{2} \right \rceil\}$. Finally, we show that for given integers
$k$ and $n$, the minimum size of a connected graph $G$ of order $n$ with
$pd(G)=k$ is $n-1$ for $k=1$ and $n+2k-4$ for $2\leq k\leq
\lceil\frac{n}{2}\rceil$.

1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
17
The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int