# Rainbow connection in graphs

Mathematica Bohemica (2008)

• Volume: 133, Issue: 1, page 85-98
• ISSN: 0862-7959

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## Abstract

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Let $G$ be a nontrivial connected graph on which is defined a coloring $c\phantom{\rule{0.222222em}{0ex}}E\left(G\right)\to \left\{1,2,...,k\right\}$, $k\in ℕ$, of the edges of $G$, where adjacent edges may be colored the same. A path $P$ in $G$ is a rainbow path if no two edges of $P$ are colored the same. The graph $G$ is rainbow-connected if $G$ contains a rainbow $u-v$ path for every two vertices $u$ and $v$ of $G$. The minimum $k$ for which there exists such a $k$-edge coloring is the rainbow connection number $\mathrm{r}c\left(G\right)$ of $G$. If for every pair $u,v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is strongly rainbow-connected. The minimum $k$ for which there exists a $k$-edge coloring of $G$ that results in a strongly rainbow-connected graph is called the strong rainbow connection number $\mathrm{s}rc\left(G\right)$ of $G$. Thus $\mathrm{r}c\left(G\right)\le \mathrm{s}rc\left(G\right)$ for every nontrivial connected graph $G$. Both $\mathrm{r}c\left(G\right)$ and $\mathrm{s}rc\left(G\right)$ are determined for all complete multipartite graphs $G$ as well as other classes of graphs. For every pair $a,b$ of integers with $a\ge 3$ and $b\ge \left(5a-6\right)/3$, it is shown that there exists a connected graph $G$ such that $\mathrm{r}c\left(G\right)=a$ and $\mathrm{s}rc\left(G\right)=b$.

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