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- W2952960427 abstract "A tree in an edge-colored graph is said to be rainbow if no two edges on the tree share the same color. An edge-coloring of $G$ is called 3-rainbow if for any three vertices in $G$, there exists a rainbow tree connecting them. The 3-rainbow index $rx_3(G)$ of $G$ is defined as the minimum number of colors that are needed in a 3-rainbow coloring of $G$. This concept, introduced by Chartrand et al., can be viewed as a generalization of the rainbow connection. In this paper, we study the 3-rainbow index by using connected three-way dominating sets and 3-dominating sets. We shown that for every connected graph $G$ on $n$ vertices with minimum degree at least $delta$ ($3leqdeltaleq5$), $rx_{3}(G)leq frac{3n}{delta+1}+4$, and the bound is tight up to an additive constant; whereas for every connected graph $G$ on $n$ vertices with minimum degree at least $delta$ ($deltageq3$), we get that $rx_{3}(G)leq nfrac{ln(delta+1)}{delta+1}(1+o_{delta}(1))+5$. In addition, we obtain some tight upper bounds of the 3-rainbow index for some special graph classes, including threshold graphs, chain graphs and interval graphs." @default.
- W2952960427 created "2019-06-27" @default.
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- W2952960427 date "2014-04-09" @default.
- W2952960427 modified "2023-09-23" @default.
- W2952960427 title "The 3-rainbow index and connected dominating sets" @default.
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- W2952960427 doi "https://doi.org/10.48550/arxiv.1404.2377" @default.
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