Strong Rainbow Connection Number Research Articles

The (Strong) Rainbow Connection Number of Join Of Ladder and Trivial Graph

Let G = (V,E) be a nontrivial, finite, and connected graph. A function c from E to {1,2,...,k},k ∈ N, can be considered as a rainbow k-coloring if every two vertices x and y in G has an x- y path. Therefore, no two path's edges receive the same color; this condition is called a “rainbow path”. The smallest positive integer k, designated by rc(G), is the G rainbow connection number. Thus, G has a rainbow k-coloring. Meanwhile, the c function is considered as a strong rainbow k-coloring within the condition for every two vertices x and y in G have an x - y rainbow path whose length is the distance between x and y. The smallest positive integer k, such as G, has a strong rainbow k-coloring; such a condition is called a strong rainbow connection number of G, denoted by src(G). In this research, the rainbow connection number and strong rainbow connection number are determined from the graph resulting from the join operation between the ladder graph and the trivial graph, denoted by rc(L_n∨K_1) and src(L_n∨K_1) respectively. So, rc (L_n∨K_1 )= src (L_n∨K_1 )=2,"for" 3≤n≤4 and rc (L_n∨K_1 )=3, while src(L_n∨K_1 )=⌈n/2⌉,"for" n≥5.

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.

On the rainbow connection number of triangle-net graph

Let G be an arbitrary non-trivial connected graph. For every two vertices u and v in G, a (u,v)-path in G is called a rainbow (u,v)-path if all edges are colored differently. Next, a rainbow (u, v)-geodesic in G is a rainbow (u,v)-path of length d(u,v). Graph G is rainbow connected if for every two vertices u,v ∈ V(G), there exists a rainbow (u,v)-path. If there exists a rainbow (u, v)-geodesic in G for every two vertices u, v ∈ V(G) then G is strongly rainbow connected. The rainbow connection number rc(G) is the minimum number of colors needed to make G rainbow connected, while the strong rainbow connection number src(G) is the minimum number of colors needed to make G strongly rainbow connected. Let Trn be the generalized triangle-ladder graph for n ≥ 2. The triangle-net graph, denoted by H = (Trn)m, is constructed by taking m homogeneous generalized triangle-ladder graphs and identifying their terminal vertices, for m ≥ 2. This paper determined the rainbow connection number of the triangle-net graph and the upper bound of the strong rainbow connection number of the graph.

Strong rainbow connection numbers of helm graphs and amalgamation of lollipop graphs

An edge coloring of a graph is called rainbow coloring if any pair of vertices are connected by a path with different colors of edges. Let G be a simple connected graph. A rainbow u — v geodesic in G is a path between two vertices u and v in G of length d(u,v), where d(u,v) is the distance of those vertices, and all of its edges are colored by different colors. If G contains a rainbow u — v geodesic for every pair of distinct vertices u and v in this graph, then G is called strongly rainbow connected graph. The strong rainbow connection number of G, denoted by src(G), is the least number of colors needed to make Gstrongly rainbow connected graph. In this paper, we prove that helm graphs and amalgamation of lollipop graphs are strongly rainbow connected graphs.

An integer program and new lower bounds for computing the strong rainbow connection numbers of graphs

AbstractWe present an integer programming model to compute the strong rainbow connection number, src(G), of any simple graph G. We introduce several enhancements to the proposed model, including a fast heuristic, and a variable elimination scheme. Moreover, we present a novel lower bound for src(G) which may be of independent research interest. We solve the integer program both directly and using an alternative method based on iterative lower bound improvement, the latter of which we show to be highly effective in practice. To our knowledge, these are the first computational methods for the strong rainbow connection problem. We demonstrate the efficacy of our methods by computing the strong rainbow connection numbers of graphs containing up to 379 vertices.

On d-local strong rainbow connection number of prism graphs

A u − ν rainbow path is a path that connects two vertices u and ν in a graph G and every edge in that path has a different color. A connected graph G is called a rainbow graph if there is a rainbow path for every pair of vertices in G. For any two vertices u and ν in G, a rainbow u − ν geodesic in G is the shortest rainbow u − ν path. The d-local strong rainbow number (lsrc d ) is the smallest number of colors needed to color the edges of G such that any two vertices with distance at most d can be connected by a rainbow geodesic. Thus, the value of d is in the interval 1 < d < diam(G). In this paper, we show the lsrc d of prism graphs with d = 2 and d = 3, and the generalized formula of lsrc d for any value of d.

Rainbow and strong rainbow connection number for some families of graphs

Let G be a nontrivial connected graph. Then G is called a rainbow connected graph if there exists a coloring c : E(G) → {1, 2, ..., k}, k ∈ N, of the edges of G, such that there is a u − v rainbow path between every two vertices of G, where a path P in G is a rainbow path if no two edges of P are colored the same. The minimum k for which there exists such a k-edge coloring is the rainbow connection number rc(G) of G. If for every pair u, v of distinct vertices, G contains a rainbow u − v geodesic, then G is called strong rainbow connected. The minimum k for which G is strong rainbow-connected is called the strong rainbow connection number src(G) of G. The exact rc and src of the rotationally symmetric graphs are determined.

Color code techniques in rainbow connection

Let G be a graph with an edge k -coloring γ : E ( G ) → {1, …, k } (not necessarily proper). A path is called a rainbow path if all of its edges have different colors. The map γ is called a rainbow coloring if any two vertices can be connected by a rainbow path. The map γ is called a strong rainbow coloring if any two vertices can be connected by a rainbow geodesic. The smallest k for which there is a rainbow k -coloring (resp. strong rainbow k -coloring) on G is called the rainbow connection number (resp. strong rainbow connection number) of G , denoted r c ( G ) (resp. s r c ( G ) ). In this paper we generalize the notion of “color codes” that was originally used by Chartrand et al. in their study of the rc and src of complete bipartite graphs, so that it now applies to any connected graph. Using color codes, we prove a new class of lower bounds depending on the existence of sets with common neighbours. Tight examples are discussed, involving the amalgamation of complete graphs, generalized wheel graphs, and a special class of sequential join of graphs.

On the Rainbow and Strong Rainbow Coloring of Comb Product Graphs

Let G=(V,E) be a simple, nontrivial, finite, connected and undirected graph. Let c be a coloring c: E(G)g{1,2,&hellip;,k}, kdN. A path of the edges of a colored graph is said to be a rainbow path if no two edges on the path have the same color but the adjacent edges may be colored by the same colors. An edge colored graph G is rainbow connected if there exists a rainbow u-v path for every two vertices u and v of G. Furthermore, for any two vertices u and v of G, a rainbow u-v geodesic in G is a rainbow u-v path of length d(u,v), where d(u,v) is the distance between u and v. The graph G is strongly rainbow connected if there exists a rainbow u-v geodesic for any two vertices u and v in G. The rainbow and strong rainbow connection numbers of a graph G, denoted by rc(G) and src(G) respectively, are the minimum number of colors that are needed in order to make G rainbow and strongly rainbow connected, respectively. Some results have shown the lower and upper bound of rc(G) and src(G), but most of them are not sharp. Thus, finding an exact value of rc(G) and src(G) are significantly useful. In this paper, we study the exact values of rainbow and strong rainbow connection numbers of comb product graphs.

Strong rainbow connection numbers of toroidal meshes

In 2011, Li et al. [The (strong) rainbow connection numbers of Cayley graphs on Abelian groups, Comput. Math. Appl. 62(11) (2011) 4082–4088] obtained an upper bound of the strong rainbow connection number of an [Formula: see text]-dimensional undirected toroidal mesh. In this paper, this bound is improved. As a result, we give a negative answer to their problem.

On rainbow connection and strong rainbow connection number of amalgamation of prism graph P3,2

Let G be a nontrivial connected graph. We follow Chartrand, et al in [2] for the definiton of rainbow connection and strong rainbow connection number. For t ∊ N and t > 2, let {P(3,2)i|i ∊ {1, 2, …, t}} is a finite collection of prism graph P3,2 that has a fixed vertex v called a terminal. The amalgamation of prism graph, Amal(P(3,2)i, v), is a graph formed by taking all the element of finite collection of prism graph P3,2 and identifying their terminal. This paper determines the rainbow connection and strong rainbow connection number from amalgamation of prism graph by a deductive reasoning method. The result are as follows: rc(Amal(P(3,2)i, v)) = 4 and src(Amal(P(3,2)i, v)) = max {4, t} for i ∊ {1, 2, …, t}.

Strong rainbow connection in digraphs

An arc-coloured digraph is strongly rainbow connected if for every pair of vertices (u,v) there exists a shortest path from u to v all of whose arcs have different colours. The strong rainbow connection number of a digraph is the minimum number of colours needed to make the graph strongly rainbow connected.In this paper, we study the strong rainbow connection number of minimally strongly connected digraphs, non-Hamiltonian strong digraphs and strong tournaments.

Total rainbow connection of digraphs

An edge-coloured path is rainbow if its edges have distinct colours. For a connected graph G, the rainbow connection number (resp. strong rainbow connection number) of G is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by a rainbow path (resp. rainbow geodesic). These two graph parameters were introduced by Chartrand, Johns, McKeon, and Zhang in 2008. Krivelevich and Yuster generalised this concept to the vertex-coloured setting. Similarly, Liu, Mestre, and Sousa introduced the version which involves total-colourings.Dorbec, Schiermeyer, Sidorowicz, and Sopena extended the concept of the rainbow connection to digraphs. In this paper, we consider the (strong) total rainbow connection number of digraphs. Results on the (strong) total rainbow connection number of biorientations of graphs, tournaments, and cactus digraphs are presented.

On the Rainbow and Strong Rainbow Connection Numbers of the m-Splitting of the Complete Graph Kn

An edge-coloring of a graph is called rainbow if any two vertices are connected by a path consisting of edges of different colors. The least number of colors in such a coloring is called the rainbow connection number of G , denoted by rc( G ). An edge-coloring of a graph is called strong rainbow if any two vertices are connected by a geodesic consisting of edges of different colors. The least number of colors in such a coloring is called the strong rainbow connection number of G , denoted by src( G ). In this paper we study the rc and src of the m -splitting of a graph. In particular we study Spl m ( K n ). We present the exact values of its rc and src in several cases, and we prove several bounds in the other cases.

Strong Rainbow Edge Coloring of Some Interconnection Networks

Abstract A rainbow edge coloring of a connected graph is a coloring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are colored the same. Rainbow coloring has received much attention recently in the field of interconnection networks. Computing the rainbow connection number of a graph is NP - hard and it finds its applications in the secure transfer of classified information between agencies and in cellular network. This paper investigates the strong rainbow connection numbers of butterfly network, Benes network and torus network.

The rainbow connection of windmill and corona graph

The rainbow connection number of G, denoted by rc(G), is the smallest number of colors needed to color its edges, so that every pair of its vertices is connected by at least one path in which no two edges are colored the same. A rainbow u v geodesic in G is a rainbow path of length d(u;v), where d(u;v) is the distance between u and v. The graph G is strongly rainbow connected if there exists a rainbow u v geodesic for any two vertices u and v in G. The strong rainbow connection number src(G) of G is the minimum number of colors needed to make G strongly rainbow connected. In this paper we determine the exact values of the windmill graph K (n) m . Moreover, we compute the rc(G H) where G or H is complete graph Km or path P2 with m is an integer. These graphs we studied have progressive connection on structure. Mathematics Subject Classication: 05C15, 05C40

Rainbow Connection and Graph Products

A path in an edge colored graph G is called a rainbow path if all its edges have pairwise different colors. Then G is rainbow connected if there exists a rainbow path between every pair of vertices of G and the least number of colors needed to obtain a rainbow connected graph is the rainbow connection number. If we demand that there must exist a shortest rainbow path between every pair of vertices, we speak about strongly rainbow connected graph and the strong rainbow connection number. In this paper we study the (strong) rainbow connection number on the direct, strong, and lexicographic product and present several upper bounds for these products that are attained by many graphs. Several exact results are also obtained.

The rainbow connection of fan and sun

A path in an edge−colored graph is said to be a rainbow path if every edge in the path has different color. An edge colored graph is rainbow connected if there exists a rainbow path between every pair of vertices. The rainbow connection of a graph G, denoted by rc(G), is the smallest number of colors required to color the edges of graph such that the graph is rainbow connected. Given two arbitrary vertices u and v in G, a rainbow u−v geodesic in G is a rainbow u−v path of length d(u,v), where d(u,v) is the distance between u and v. The graph G is strongly rainbow connected if there exists a rainbow u−v geodesic for any two vertices u and v in G. The strong rainbow connection number of G, denoted by src(G), is the minimum number of colors needed to make G strongly rainbow connected. In this paper we determine the exact values of rc(G) and src(G) where G are Fn and Sm with n + 1 and 2m vertices, respectively.

The (strong) rainbow connection numbers of Cayley graphs on Abelian groups

A path in an edge-colored graph G , whose 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 r c ( 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. The strong rainbow connection number s r c ( G ) of G is the minimum integer i for which there exists an i -edge-coloring of G such that every two distinct vertices u and v of G are connected by a rainbow path of length d ( u , v ) . In this paper, we give upper and lower bounds of the (strong) rainbow connection numbers of Cayley graphs on Abelian groups. Moreover, we determine the (strong) rainbow connection numbers of some special cases.

Rainbow connection in graphs

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 $\mathop {\mathrm src}(G)=b$.