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

Note on minimally kk-rainbow connected graphs

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can use $k$ colors to make $G$ rainbow connected. For integers $n$ and $d$ let $t(n,d)$ denote the minimum size (number of edges) in $k$-rainbow connected graphs of order $n$. Schiermeyer got some exact values and upper bounds for $t(n,d)$. However, he did not get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$. In this paper, we improve his lower bound of $t(n,2)$, and get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$.Comment: 8 page

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

Algorithms and Bounds for Very Strong Rainbow Coloring

A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (\src(G)) of the graph. When proving upper bounds on \src(G), it is natural to prove that a coloring exists where, for \emph{every} shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call \emph{very strong rainbow connection number} (\vsrc(G)). In this paper, we give upper bounds on \vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on \src(G) for these classes, showing that the study of \vsrc(G) enables meaningful progress on bounding \src(G). Then we study the complexity of the problem to compute \vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that \vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for \src(G). We also observe that deciding whether \vsrc(G) = k is fixed-parameter tractable in $k$ and the treewidth of $G$. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor $n^{1-\varepsilon}$, unless P$=$NP

Meander, Folding and Arch Statistics

The statistics of meander and related problems are studied as particular realizations of compact polymer chain foldings. This paper presents a general discussion of these topics, with a particular emphasis on three points: (i) the use of a direct recursive relation for building (semi) meanders (ii) the equivalence with a random matrix model (iii) the exact solution of simpler related problems, such as arch configurations or irreducible meanders.Comment: 82 pages, uuencoded, uses harvmac (l mode) and epsf, 26+7 figures include