Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs

A path in an edge-colored graph $G$ is rainbow if no two edges of it are colored the same. The graph $G$ is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph $G$ is strongly rainbow-connected. The minimum number of colors needed to make $G$ rainbow-connected is known as the rainbow connection number of $G$, and is denoted by $\text{rc}(G)$. Similarly, the minimum number of colors needed to make $G$ strongly rainbow-connected is known as the strong rainbow connection number of $G$, and is denoted by $\text{src}(G)$. We prove that for every $k \geq 3$, deciding whether $\text{src}(G) \leq k$ is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an $n$-vertex split graph with a factor of $n^{1/2-\epsilon}$ for any $\epsilon > 0$ unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.Comment: 13 pages, 3 figure

On Rainbow Connection Number and Connectivity

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 vertices is connected by at least one path in which no two edges are coloured the same. In this paper we investigate the relationship of rainbow connection number with vertex and edge connectivity. It is already known that for a connected graph with minimum degree $\delta$, the rainbow connection number is upper bounded by $3n/(\delta + 1) + 3$ [Chandran et al., 2010]. This directly gives an upper bound of $3n/(\lambda + 1) + 3$ and $3n/(\kappa + 1) + 3$ for rainbow connection number where $\lambda$ and $\kappa$, respectively, denote the edge and vertex connectivity of the graph. We show that the above bound in terms of edge connectivity is tight up-to additive constants and show that the bound in terms of vertex connectivity can be improved to $(2 + \epsilon)n/\kappa + 23/ \epsilon^2$, for any $\epsilon > 0$. We conjecture that rainbow connection number is upper bounded by $n/\kappa + O(1)$ and show that it is true for $\kappa = 2$. We also show that the conjecture is true for chordal graphs and graphs of girth at least 7.Comment: 10 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

On Local-Strong Rainbow Connection Numbers On Generalized Prism Graphs And Generalized Antiprism Graphs

Rainbow geodesic is the shortest path that connects two different vertices in graph  such that every edge of the path has different colors. The strong rainbow connection number of a graph G, denoted by src(G), is the smallest number of colors required to color the edges of G such that there is a rainbow geodesic for each pair of vertices. The d-local strong rainbow connection number, denoted by lrscd, is the smallest number of colors required to color the edges of G such that any pair of vertices with a maximum distance d is connected by a rainbow geodesic. This paper contains some results of lrscd of  generalized prism graphs (PmxCn) and generalized antiprism graphs  for values of d=2, d=3, and d=4

Rainbow Connection Number of Prism and Product of Two Graphs

An edge-colouring of a graph $G$ is rainbow connected if, for any two vertices of $G$, there are $k$ internally vertex-disjoint paths joining them, each of which is rainbow and then a minimal numbers of color $G$ is required to make rainbow connected. The rainbow connection numbers of a connected graph $G$, denoted $rc(G)$. In this paper we will discuss the rainbow connection number $rc(G)$ for some special graphs and its operations, namely prism graph $P_{m,n}$, antiprism graph $AP_{n}$, tensor product of $C_{3}$ $\bigotimes$ $L_{n}$, joint graph $\bar{K_{3}}$+$C_{n}$

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

Total Rainbow Connection Number Of Shackle Product Of Antiprism Graph (〖AP〗_3)

Function if  is said to be k total rainbows in , for each pair of vertex  there is a path called  with each edge and each vertex on the path will have a different color. The total connection number is denoted by trc  defined as the minimum number of colors needed to make graph  to be total rainbow connected. Total rainbow connection numbers can also be applied to graphs that are the result of operations. The denoted shackle graph  is a graph resulting from the denoted graph  where t is number of copies of G. This research discusses rainbow connection numbers rc and total rainbow connection trc(G) using the shackle operation, where  is the antiprism graph . Based on this research, rainbow connection numbers rc shack , and total rainbow connection trc shack for .Fungsi jika c : G → {1,2,. . . , k} dikatakan k total pelangi pada G, untuk setiap pasang titik &nbsp;terdapat lintasan disebut x-y dengan setiap sisi dan setiap titik pada lintasan akan memiliki warna berbeda. Bilangan terhubung total pelangi dilambangkan dengan trc(G), didefinisikan sebagai jumlah minimum warna yang diperlukan untuk membuat graf G menjadi terhubung-total pelangi. Bilangan terhubung total pelangi juga dapat diterapkan pada graf yang merupakan hasil operasi. Graf shackle yang dilambangkan (G1,G2,…,Gt) adalah graf yang dihasilkan dari graf G yang dilambangkan (G,t) dengan t adalah jumlah salinan dari &nbsp;Penelitian ini membahas mengenai bilangan terhubung pelangi rc&nbsp;dan bilangan terhubung total pelangi trc(G)menggunakan operasi shackle, dimana G adalah graf Antiprisma (AP3)Berdasarkan penelitian ini, diperoleh bilangan terhubung pelangi rc(shack AP3,t )= t+2, dan total pelangi trc(shack AP3,t)=2t+3 untuk t ≥2