Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On (strong) proper vertex-connection of graphs   A path in a vertex-colored graph is a {\it vertex-proper path} if any two
internal adjacent vertices differ in color. A vertex-colored graph is {\it
proper vertex $k$-connected} if any two vertices of the graph are connected by
$k$ disjoint vertex-proper paths of the graph. For a $k$-connected graph $G$,
the {\it proper vertex $k$-connection number} of $G$, denoted by $pvc_{k}(G)$,
is defined as the smallest number of colors required to make $G$ proper vertex
$k$-connected. A vertex-colored graph is {\it strong proper vertex-connected},
if for any two vertices $u,v$ of the graph, there exists a vertex-proper
$u$-$v$ geodesic. For a connected graph $G$, the {\it strong proper
vertex-connection number} of $G$, denoted by $spvc(G)$, is the smallest number
of colors required to make $G$ strong proper vertex-connected. These concepts
are inspired by the concepts of rainbow vertex $k$-connection number
$rvc_k(G)$, strong rainbow vertex-connection number $srvc(G)$, and proper
$k$-connection number $pc_k(G)$ of a $k$-connected graph $G$. Firstly, we
determine the value of $pvc(G)$ for general graphs and $pvc_k(G)$ for some
specific graphs. We also compare the values of $pvc_k(G)$ and $pc_k(G)$. Then,
sharp bounds of $spvc(G)$ are given for a connected graph $G$ of order $n$,
that is, $0\leq spvc(G)\leq n-2$. Moreover, we characterize the graphs of order
$n$ such that $spvc(G)=n-2,n-3$, respectively. Finally, we study the
relationship among the three vertex-coloring parameters, namely, $spvc(G), \
srvc(G)$ and the chromatic number $\chi(G)$ of a connected graph $G$.

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The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys