Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nordhaus-Gaddum-type theorem for total proper connection number of
  graphs   A graph is said to be \emph{total-colored} if all the edges and the vertices
of the graph are colored. A path $P$ in a total-colored graph $G$ is called a
\emph{total-proper path} if $(i)$ any two adjacent edges of $P$ are assigned
distinct colors; $(ii)$ any two adjacent internal vertices of $P$ are assigned
distinct colors; $(iii)$ any internal vertex of $P$ is assigned a distinct
color from its incident edges of $P$. The total-colored graph $G$ is
\emph{total-proper connected} if any two distinct vertices of $G$ are connected
by a total-proper path. The \emph{total-proper connection number} of a
connected graph $G$, denoted by $tpc(G)$, is the minimum number of colors that
are required to make $G$ total-proper connected. In this paper, we first
characterize the graphs $G$ on $n$ vertices with $tpc(G)=n-1$. Based on this,
we obtain a Nordhaus-Gaddum-type result for total-proper connection number. We
prove that if $G$ and $\overline{G}$ are connected complementary graphs on $n$
vertices, then $6\leq tpc(G)+tpc(\overline{G})\leq n+2$. Examples are given to
show that the lower bound is sharp for $n\geq 4$. The upper bound is reached
for $n\geq 5$ if and only if $G$ or $\overline{G}$ is the tree with maximum
degree $n-2$.

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The categories of this paper are: nlin
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: supr-con
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat