scispace - formally typeset
Search or ask a question
Posted Content

Rainbow Connection Number and Radius

TL;DR: Chakraborty et al. as discussed by the authors showed that for any bridgeless graph G with radius r, rc(G) <= r(r + 2) and showed that this bound is the best possible for any graph G as a function of r, not just for bridgless graphs, but also for graphs of any stronger connectivity.
Abstract: 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.
Citations
More filters
Posted Content
TL;DR: The concept of Rainbow Connection was introduced by Chartrand et al. in 2008 as discussed by the authors, and quite a lot papers have been published about it, and a survey of the results and papers that dealt with it can be found here.
Abstract: The concept of rainbow connection was introduced by Chartrand et al. in 2008. It is fairly interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow $k$-connectivity, $k$-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems or questions.

223 citations

Journal ArticleDOI
18 Aug 2017
TL;DR: This survey attempts to bring together most of the new results and papers that deal with the concept of rainbow connection in graph theory, and tries to organize the work into the following categories.
Abstract: The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow k-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study.

54 citations

Posted Content
TL;DR: It is shown that rc(G)@?5 if G is a bridgeless graph with diameter 2, and that rc (G) @?k+2 ifG is a connected graph withiameter 2 and has k bridges, where k>=1.
Abstract: A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. It is known that for a graph $G$ with diameter 2, to determine $rc(G)$ is NP-hard. So, it is interesting to know the best upper bound of $rc(G)$ for such a graph $G$. In this paper, we show that $rc(G)\leq 5$ if $G$ is a bridgeless graph with diameter 2, and that $rc(G)\leq k+2$ if $G$ is a connected graph of diameter 2 with $k$ bridges, where $k\geq 1$.

38 citations

Journal ArticleDOI
TL;DR: It is proved that with probability tending to one as $n$ goes to infinity the rainbow connection of G satisfies $rc(G)=O(\log n)$, which is best possible up to a hidden constant.
Abstract: An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this work we study the rainbow connection of the random $r$-regular graph $G=G(n,r)$ of order $n$, where $r\ge 4$ is a constant. We prove that with probability tending to one as $n$ goes to infinity the rainbow connection of $G$ satisfies $rc(G)=O(\log n)$, which is best possible up to a hidden constant.

36 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that for a graph G with diameter 2, the problem of computing the Rainbow Connection Number (RCN) is NP-hard and that rc(G)@?k+2 is the minimum integer k for which any two distinct vertices of G are connected.

26 citations

References
More filters
Journal ArticleDOI
01 Jan 2008
TL;DR: In this article, it was shown that the strong rainbow connection number (SRC) is the minimum number of edges of a graph for which there exists a strongly rainbow-connected edge coloring of the graph.
Abstract: Let $G$ be a nontrivial connected graph on which is defined a coloring $c\: E(G) \rightarrow \lbrace 1, 2, \ldots , k\rbrace $, $k \in {\mathbb{N}}$, 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 $\mathop {\mathrm rc}(G)$ 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 $\mathop {\mathrm src}(G)$ of $G$. Thus $\mathop {\mathrm rc}(G) \le \mathop {\mathrm src}(G)$ for every nontrivial connected graph $G$. Both $\mathop {\mathrm rc}(G)$ and $\mathop {\mathrm src}(G)$ 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 $b \ge (5a-6)/3$, it is shown that there exists a connected graph $G$ such that $\mathop {\mathrm rc}(G)=a$ and $\mathop {\mathrm src}(G)=b$.

523 citations

Book
22 Sep 2008
TL;DR: Chromatic Graph Theory as discussed by the authors explores connections between major topics in graph theory and graph colorings as well as emerging topics, including trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings.
Abstract: Beginning with the origin of the four color problem in 1852, the field of graph colorings has developed into one of the most popular areas of graph theory. Introducing graph theory with a coloring theme, Chromatic Graph Theory explores connections between major topics in graph theory and graph colorings as well as emerging topics. This self-contained book first presents various fundamentals of graph theory that lie outside of graph colorings, including basic terminology and results, trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. The remainder of the text deals exclusively with graph colorings. It covers vertex colorings and bounds for the chromatic number, vertex colorings of graphs embedded on surfaces, and a variety of restricted vertex colorings. The authors also describe edge colorings, monochromatic and rainbow edge colorings, complete vertex colorings, several distinguishing vertex and edge colorings, and many distance-related vertex colorings. With historical, applied, and algorithmic discussions, this text offers a solid introduction to one of the most popular areas of graph theory.

303 citations

Journal ArticleDOI
TL;DR: This note presents ent algorithm which uses any search method to And all the bridges of a graph, its maxin% connected (bridgeconnected) szbgraphs.

224 citations

Journal ArticleDOI
TL;DR: This paper proves several non-trivial upper bounds for $rc(G)$, as well as determine sufficient conditions that guarantee that if $G$ is a connected graph with $n$ vertices and with minimum degree $3$ then $rc (G)=2$.
Abstract: An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper we prove several non-trivial upper bounds for $rc(G)$, as well as determine sufficient conditions that guarantee $rc(G)=2$. Among our results we prove that if $G$ is a connected graph with $n$ vertices and with minimum degree $3$ then $rc(G)

219 citations

Journal IssueDOI
TL;DR: In this paper, Caro et al. showed that if G has n vertices and minimum degree δ then rvc(G) <20n-δ, which is the smallest number of colors that are needed in order to make Grainbow edge-connected.
Abstract: An edge-colored graph Gis rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make Grainbow edge-connected. We prove that if Ghas nvertices and minimum degree δ then rc(G)<20n-δ. This solves open problems from Y. Caro, A. Lev, Y. Roditty, Z. Tuza, and R. Yuster (Electron J Combin 15 (2008), nR57) and S. Chakrborty, E. Fischer, A. Matsliah, and R. Yuster (Hardness and algorithms for rainbow connectivity, Freiburg (2009), pp. 243–254). A vertex-colored graph Gis rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make Grainbow vertex-connected. One cannot upper-bound one of these parameters in terms of the other. Nevertheless, we prove that if Ghas nvertices and minimum degree δ then rvc(G)<11n-δ. We note that the proof in this case is different from the proof for the edge-colored case, and we cannot deduce one from the other. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 185–191, 2010

212 citations