Rainbow Connection Number and Connected Dominating Sets

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2, where {\gamma}_c(G) is the connected domination number of G. Bounds of the form diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds. An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree {\delta}, the rainbow connection number is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of Schiermeyer (2009), improving the previously best known bound of 20n/{\delta} by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up to additive factors by a construction of Caro et al. (2008).Comment: 14 page

Note on the upper bound of the rainbow index of a graph

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 $2\leq k\leq n$. 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 $G$.Comment: 12 pages. arXiv admin note: text overlap with arXiv:0902.1255 by other author

Rainbow connection number, bridges and radius

Let $G$ be a connected graph. The notion \emph{the rainbow connection number $rc(G)$} of a graph $G$ was introduced recently by Chartrand et al. Basavaraju et al. showed that for every bridgeless graph $G$ with radius $r$, $rc(G)\leq r(r+2)$, and the bound is tight. In this paper, we prove that if $G$ is a connected graph, and $D^{k}$ is a connected $k$-step dominating set of $G$, then $G$ has a connected $(k-1)$-step dominating set $D^{k-1}\supset D^{k}$ such that $rc(G[D^{k-1}])\leq rc(G[D^{k}])+\max\{2k+1,b_k\}$, where $b_k$ is the number of bridges in $E(D^{k}, N(D^{k}))$. Furthermore, for a connected graph $G$ with radius $r$, let $u$ be the center of $G$, and $D^{r}=\{u\}$. Then $G$ has $r-1$ connected dominating sets $D^{r-1}, D^{r-2},..., D^{1}$ satisfying $D^{r}\subset D^{r-1}\subset D^{r-2} ...\subset D^{1}\subset D^{0}=V(G)$, and $rc(G)\leq \sum_{i=1}^{r}\max\{2i+1,b_i\}$, where $b_i$ is the number of bridges in $E(D^{i}, N(D^{i})), 1\leq i \leq r$. From the result, we can get that if for all $1\leq i\leq r, b_i\leq 2i+1$, then $rc(G)\leq \sum_{i=1}^{r}(2i+1)= r(r+2)$; if for all $1\leq i\leq r, b_i> 2i+1$, then $rc(G)= \sum_{i=1}^{r}b_i$, the number of bridges of $G$. This generalizes the result of Basavaraju et al.Comment: 8 page

Rainbow Connection Number and Radius

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) <= r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (Star graph for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) <= rk. It is known that computing rc(G) is NP-Hard [Chakraborty et al., 2009]. Here, we present a (r+3)-factor approximation algorithm which runs in O(nm) time and a (d+3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.Comment: Revised preprint with an extra section on an approximation algorithm. arXiv admin note: text overlap with arXiv:1101.574