Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Total proper connection of graphs   A graph is said to be {\it total-colored} if all the edges and the vertices
of the graph is colored. A path in a total-colored graph is a {\it total proper
path} if $(i)$ any two adjacent edges on the path differ in color, $(ii)$ any
two internal adjacent vertices on the path differ in color, and $(iii)$ any
internal vertex of the path differs in color from its incident edges on the
path. A total-colored graph is called {\it total-proper connected} if any two
vertices of the graph are connected by a total proper path of the graph. For a
connected graph $G$, the {\it total proper connection number} of $G$, denoted
by $tpc(G)$, is defined as the smallest number of colors required to make $G$
total-proper connected. These concepts are inspired by the concepts of proper
connection number $pc(G)$, proper vertex connection number $pvc(G)$ and total
rainbow connection number $trc(G)$ of a connected graph $G$. In this paper, we
first determine the value of the total proper connection number $tpc(G)$ for
some special graphs $G$. Secondly, we obtain that $tpc(G)\leq 4$ for any
$2$-connected graph $G$ and give examples to show that the upper bound $4$ is
sharp. For general graphs, we also obtain an upper bound for $tpc(G)$.
Furthermore, we prove that $tpc(G)\leq \frac{3n}{\delta+1}+1$ for a connected
graph $G$ with order $n$ and minimum degree $\delta$. Finally, we compare
$tpc(G)$ with $pvc(G)$ and $pc(G)$, respectively, and obtain that
$tpc(G)>pvc(G)$ for any nontrivial connected graph $G$, and that $tpc(G)$ and
$pc(G)$ can differ by $t$ for $0\leq t\leq 2$.

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The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
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The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
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The categories of this paper are: nlin