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- W2952938233 abstract "A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is a rainbow path if every two edges of it receive distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the minimum number of colors that are needed to color the edges of $G$ such that there exists a rainbow path connecting every two vertices of $G$. Similarly, a tree in $G$ is a rainbow~tree if no two edges of it receive the same color. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow tree connecting $S$ for each $k$-subset $S$ of $V(G)$ is called the $k$-rainbow index of $G$, denoted by $rx_k(G)$, where $k$ is an integer such that $2leq kleq n$. Chakraborty et al. got the following result: For every $epsilon> 0$, a connected graph with minimum degree at least $epsilon n$ has bounded rainbow connection, where the bound depends only on $epsilon$. Krivelevich and Yuster proved that if $G$ has $n$ vertices and the minimum degree $delta(G)$ then $rc(G)<20n/delta(G)$. This bound was later improved to $3n/(delta(G)+1)+3$ by Chandran et al. Since $rc(G)=rx_2(G)$, a natural problem arises: for a general $k$ determining the true behavior of $rx_k(G)$ as a function of the minimum degree $delta(G)$. In this paper, we give upper bounds of $rx_k(G)$ in terms of the minimum degree $delta(G)$ in different ways, namely, via Szemer'{e}di's Regularity Lemma, connected $2$-step dominating sets, connected $(k-1)$-dominating sets and $k$-dominating sets of $G$." @default.
- W2952938233 created "2019-06-27" @default.
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- W2952938233 date "2014-07-17" @default.
- W2952938233 modified "2023-10-03" @default.
- W2952938233 title "Note on the upper bound of the rainbow index of a graph" @default.
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- W2952938233 doi "https://doi.org/10.48550/arxiv.1407.4663" @default.
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