Rainbow Connection Number Of Graph Research Articles

Proper (Strong) Rainbow Connection and Proper (Strong) Rainbow Vertex Connection of Graphs with Large Clique Number

The proper rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] to make [Formula: see text] rainbow vertex connected. The proper strong rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] to make [Formula: see text] strong rainbow vertex connected. These two concepts are inspired by the concepts of proper (strong) rainbow connection number of graphs. In this paper, we determine the values of [Formula: see text] and [Formula: see text] of [Formula: see text] with large clique numbers [Formula: see text] and [Formula: see text]. Moreover, we determine the values of [Formula: see text] and [Formula: see text] of [Formula: see text] with large clique numbers [Formula: see text], [Formula: see text] and [Formula: see text].

On the Locating Rainbow Connection Number of Trees and Regular Bipartite Graphs

Locating the rainbow connection number of graphs is a new mathematical concept that combines the concepts of the rainbow vertex coloring and the partition dimension. In this research, we determine the lower and upper bounds of the locating rainbow connection number of a graph and provide the characterization of graphs with the locating rainbow connection number equal to its upper and lower bounds to restrict the upper and lower bounds of the locating rainbow connection number of a graph. We also found the locating rainbow connection number of trees and regular bipartite graphs. The method used in this study is a deductive method that begins with a literature study related to relevant previous research concepts and results, making hypotheses, conducting proofs, and drawing conclusions. This research concludes that only path graphs with orders 2, 3, 4, and complete graphs have a locating rainbow connection number equal to 2 and the order of graph G, respectively. We also showed that the locating rainbow connection number of bipartite regular graphs is in the range of r-⌊n/4⌋+2 to n/2+1, and the locating rainbow connection number of a tree is determined based on the maximum number of pendants or the maximum number of internal vertices. Doi: 10.28991/ESJ-2023-07-04-016 Full Text: PDF

On the locating rainbow connection number of amalgamation of complete graphs

Locating rainbow connection number determines the minimum number of colors connecting any two vertices of a graph with a rainbow vertex path and also verifies that the given colors produce a different rainbow code for each vertex. Locating rainbow connection number of graphs is a new mathematical concept, especially in graph theory, which combines the concepts of the rainbow vertex coloring and the partition dimension. In this paper, we determine the locating rainbow connection number of amalgamation of complete graphs.

Proper (Strong) Rainbow Connection and Proper (Strong) Rainbow Vertex Connection of Some Special Graphs

The proper rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is rainbow vertex connected. The proper strong rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is strong rainbow vertex connected. These two concepts are inspired by the concept of proper (strong) rainbow connection number of graphs. In this paper, we first determine the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], pencil graphs, circular ladders or Möbius ladders. Secondly, we obtain the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], paths, cycles, wheels, complete multipartite graphs, pencil graphs, circular ladders and Möbius ladders. Finally, we characterize all the connected graphs [Formula: see text] with [Formula: see text] and [Formula: see text].

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.

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.

Proper rainbow connection number of graphs

A path in an edge-coloured graph is called \emph{rainbow path} if its edges receive pairwise distinct colours. An edge-coloured graph is said to be \emph{rainbow connected} if any two distinct vertices of the graph are connected by a rainbow path. The minimum $k$ for which there exists such an edge-colouring is the rainbow connection number $rc(G)$ of $G.$ Recently, Bau et al. \cite{BJJKM2018} introduced this concept with the additional requirement that the edge-colouring must be proper. %An proper edge-coloured graph is said to be \emph{properly rainbow connected} if any two distinct vertices of the graph are connected by a rainbow path. The \emph{proper rainbow connection number} of $G$, denoted by $prc(G)$, is the minimum number of colours needed in order to make it properly rainbow connected. In this paper we first prove an improved upper bound $prc(G) \leq n$ for every connected graph $G$ of order $n \geq 3.$ Next we show that the difference $prc(G) - rc(G)$ can be arbitrarily large. Finally, we present several sufficient conditions for graph classes satisfying $prc(G) = \chi'(G).$

BILANGAN KETERHUBUNGAN PELANGI PADA PEWARNAAN-SISI GRAF

Let be a graph. An edge-coloring of is a function , where is a set of colors. Respect to a subgraph of is called a rainbow subgraph if all edges of get different colors. Graph is called rainbow connected if for every two distinct vertices of is joined by a rainbow path. The rainbow connection number of , denoted by , is the minimum number of colors needed in coloring all edges of such that is a rainbow connected. The main problem considered in this thesis is determining the rainbow connection number of graph. In this thesis, we determine the exact value of the rainbow connection number of some classes of graphs such as Cycles, Complete graph, and Tree. We also determining the lower bound and upper bound for the rainbow connection number of graph.
 Keywords: Rainbow Connection Number, Graph, Edge-Coloring on Graph.

Total rainbow connection numbers of some special graphs

In 2008, Chartrand et al. first introduced the concept of rainbow connection. Since then the study of rainbow connection has received considerable attention in the literature, and now it becomes an active topic in graph theory. As a natural generalization, Uchizawa et al. (2013) and Liu et al. (2014) presented the concept of total rainbow connection, respectively. In this paper, we investigate the total rainbow connection numbers of outerplanar graphs with diameter 2. Applying our result, we improve the main result of [X. Huang, X. Li, Y. Shi, J. Yue, Y. Zhao, Rainbow connections for outerplanar graphs with diameter 2 and 3, Applied Mathematics and Computation, 242(2014), 277–280]. Next, we revise the main result of [Y. Liu, Z. Wang, Rainbow Connection Number of the Thorn Graph, Applied Mathematical Sciences, 8(2014), 6373–6377], and determine the total rainbow connection numbers of graphs G, where G are the thorn graph of complete graph Kn*, the thorn graph of the cycle Cn*. At last, we study the rainbow 2-connection numbers of some special 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,…,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.

Graphs with small total rainbow connection number

A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that $${\text{trc(G) = 3 if}}\left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) + 1 \leqslant \left| {{\text{E(G)}}} \right| \leqslant \left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right) - 1$$ , and $${\text{trc(G)}} \leqslant {\text{6 if }}\left| {{\text{E(G)}}} \right| \geqslant \left( {\begin{array}{*{20}{c}} {n - 2} \\ 2 \end{array}} \right) + 2$$ . Next, we investigate the total rainbow connection numbers of graphs G with |V(G)| = n, diam(G) ≥ 2, and clique number ω(G) = n − s for 1 ≤ s ≤ 3. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313–320] is not completely correct, and we provide a complete result for this theorem.

On Rainbow k-Connection Number of Special Graphs and It’s Sharp Lower Bound

Let G = (V, E) be a simple, nontrivial, finite, connected and undirected graph. Let c be a coloring c : E(G) → {1, 2, …, s}, s ∈ N. A path of edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph G is said to be a rainbow connected graph if there exists a rainbow u − v path for every two vertices u and v of G. The rainbow connection number of a graph G, denoted by rc(G), is the smallest number of k colors required to edge color the graph such that the graph is rainbow connected. Furthermore, for an l-connected graph G and an integer k with 1 ≤ k ≤ l, the rainbow k-connection number rck(G) of G is defined to be the minimum number of colors required to color the edges of G such that every two distinct vertices of G are connected by at least k internally disjoint rainbow paths. In this paper, we determine the exact values of rainbow connection number of some special graphs and obtain a sharp lower bound.

Rainbow connection number of graphs with diameter 3

Rainbow connection number of graphs with diameter 3

Graphs with vertex rainbow connection number two

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 graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. A vertex colored graph G is vertex rainbow connected if any two vertices are connected by a path whose internal vertices have distinct colors. The vertex rainbow connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G vertex rainbow connected. In 2011, Kemnitz and Schiermeyer considered graphs with rc(G) = 2. We investigate graphs with rvc(G) = 2. First, we prove that rvc(G) ⩽ 2 if \(\left| {E(G)} \right| \geqslant \left( {\begin{array}{*{20}c} {n - 2} \\ 2 \\ \end{array} } \right) + 2\), and the bound is sharp. Denote by s(n, 2) the minimum number such that, for each graph G of order n, we have rvc(G) ⩽ 2 provided |E(G)| ⩾ s(n, 2). It is proved that \(s(n,2) = \left( {\begin{array}{*{20}c} {n - 2} \\ 2 \\ \end{array} } \right) + 2\). Next, we characterize the vertex rainbow connection numbers of graphs G with |V (G)| = n, diam(G) ⩾ 3 and clique number ω(G) = n−s for 1 ⩽ s ⩽ 4.

Rainbow connections for outerplanar graphs with diameter 2 and 3

An edge-colored graph G is rainbow connected if every two vertices are connected by a path whose edges have distinct colors. The rainbow connection number 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. It was proved that computing rc(G) is an NP-hard problem, as well as that even deciding whether a graph has rc(G)=2 is NP-complete. Li et al. proved that rc(G)⩽5 if G is a bridgeless graph with diameter 2, while rc(G)⩽9 if G is a bridgeless graph with diameter 3. Furthermore, Uchizawa et al. showed that determining the rainbow connection number of graphs is strongly NP-complete even for outerplanar graphs. In this paper, we give upper bounds of the rainbow connection number of outerplanar graphs with small diameters.

Rainbow Connection Number of Graph Power and Graph Products

Rainbow connection number, rc(G), of a connected graph G is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same (note that the coloring need not be proper). In this paper we study the rainbow connection number with respect to three important graph product operations (namely the Cartesian product, the lexicographic product and the strong product) and the operation of taking the power of a graph. In this direction, we show that if G is a graph obtained by applying any of the operations mentioned above on non-trivial graphs, then rc(G) ≤ 2r(G) + c, where r(G) denotes the radius of G and $${c \in \{0, 1, 2\}}$$ . In general the rainbow connection number of a bridgeless graph can be as high as the square of its radius [1]. This is an attempt to identify some graph classes which have rainbow connection number very close to the obvious lower bound of diameter (and thus the radius). The bounds reported are tight up to additive constants. The proofs are constructive and hence yield polynomial time $${(2 + \frac{2}{r(G)})}$$ -factor approximation algorithms.

The rainbow connection number of 2-connected graphs

The rainbow connection number of a graph G is the least number of colours in a (not necessarily proper) edge-colouring of G such that every two vertices are joined by a path which contains no colour twice. Improving a result of Caro et al., we prove that the rainbow connection number of every 2-connected graph with n vertices is at most ⌈n/2⌉. The bound is optimal.

Bounds for the rainbow connection number of graphs

Bounds for the rainbow connection number of graphs