Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Good upper bounds for the total rainbow connection of graphs   A total-colored graph is a graph $G$ such that both all edges and all
vertices of $G$ are colored. A path in a total-colored graph $G$ is a total
rainbow path if its edges and internal vertices have distinct colors. A
total-colored graph $G$ is total-rainbow connected if any two vertices of $G$
are connected by a total rainbow path of $G$. The total rainbow connection
number of $G$, denoted by $trc(G)$, is defined as the smallest number of colors
that are needed to make $G$ total-rainbow connected. These concepts were
introduced by Liu et al. Notice that for a connected graph $G$, $2diam(G)-1\leq
trc(G)\leq 2n-3$, where $diam(G)$ denotes the diameter of $G$ and $n$ is the
order of $G$. In this paper we show, for a connected graph $G$ of order $n$
with minimum degree $\delta$, that $trc(G)\leq6n/{(\delta+1)}+28$ for
$\delta\geq\sqrt{n-2}-1$ and $n\geq 291$, while $trc(G)\leq7n/{(\delta+1)}+32$
for $16\leq\delta\leq\sqrt{n-2}-2$ and
$trc(G)\leq7n/{(\delta+1)}+4C(\delta)+12$ for $6\leq\delta\leq15$, where
$C(\delta)=e^{\frac{3\log({\delta}^3+2{\delta}^2+3)-3(\log3-1)}{\delta-3}}-2$.
This implies that when $\delta$ is in linear with $n$, then the total rainbow
number $trc(G)$ is a constant. We also show that $trc(G)\leq 7n/4-3$ for
$\delta=3$, $trc(G)\leq8n/5-13/5$ for $\delta=4$ and $trc(G)\leq3n/2-3$ for
$\delta=5$. Furthermore, an example shows that our bound can be seen tight up
to additive factors when $\delta\geq\sqrt{n-2}-1$.

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The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: cs
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex
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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: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga