Rainbow Connection Research Articles

On Rainbow Connection Number of Some Graphs

The Rainbow connection number for the following graphs, two copies of Fan graph by a path , Arrow graph and Θ , Jellyfish graph and Cycle Cactus graph have been described in this paper

Rainbow Connection Number on Amalgamation of General Prism Graph

AbstractLet be a nontrivial connected graph, the rainbow-k-coloring of graph G is the mapping of c: E(G)-> {1,2,3,…,k} such that any two vertices from the graph can be connected by a rainbow path (the path with all edges of different colors). The least natural number

Monochromatic and rainbow 4-connectivity of moon graphs

Monochromatic and rainbow connectivity are known as the concepts that can calculate the minimum number of passwords and security systems. In this paper, a security system is represented by a special mathematics object, namely a graph. The study focuses on the moon graphs or Mon. Moon graphs are chosen because it is one of the 4-connected graphs that has a minimum number of edges. The research method used is mathematical modeling equipped with direct proves and contradictions. The conclusion is that there is an efficiency of 75% when using the monochromatic connected number, and there is an efficiency between 62.5% - 75% when using the rainbow 4-connected number.

Internalized Homophobia, Depressive Symptoms, and Suicidal Ideation Among Lesbian, Gay, and Bisexual Adults in South Korea: An Age-Stratified Analysis.

Purpose: Little research on internalized homophobia (IHP) and mental health among lesbian, gay, and bisexual (LGB) individuals has been done in South Korea, a context in which LGB populations widely experience societal stigma. This study examined the association between IHP and depressive symptoms and suicidal ideation among Korean LGB adults and investigated whether the association differed by participants' age. Methods: Cross-sectional data from the "Rainbow Connection Project I - Korean Lesbian, Gay, & Bisexual Adults' Health Study" were used for this study (N = 2178). Descriptive analyses were conducted to explore the distribution of IHP, depressive symptoms, and suicidal ideation across all covariates. Multivariate analyses were also used to investigate the overall and age-stratified associations between IHP and depressive symptoms and suicidal ideation. Results: The prevalence of high IHP was greater among older LGB adults, whereas the prevalence of depressive symptoms and suicidal ideation was higher among younger LGB adults. LGB individuals with high IHP levels had significantly greater risk of depressive symptoms than those with low IHP levels. In the age-stratified analyses, the associations between IHP and depressive symptoms were statistically significant only among older LGB adults with high IHP levels. Regarding suicidal ideation, the overall association was statistically significant in the full sample, but not in the age-stratified analyses. Conclusion: These findings suggest that mental health interventions are needed for LGB adults who have high IHP levels, especially among older generations. Furthermore, greater efforts are needed to enact protective legislation for sexual minority individuals in South Korea.

The Analysis of Student Metacognition Skill in Solving Rainbow Connection Problem under the Implementation of Research-Based Learning Model

The Analysis of Student Metacognition Skill in Solving Rainbow Connection Problem under the Implementation of Research-Based Learning Model

The rainbow connectivity of cartesian product graphs

An edge-coloured graph G is said to be rainbow-connected if any two vertices are connected by a path whose edges have different colours. The rainbow connection number of a graph is the minimum number of colours needed to make the graph rainbow-connected. This parameter was introduced by G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang in 2008. Similar to rainbow connection coloring, an edge-coloring is a rainbow k-connection coloring if there are at least k internally disjoint rainbow u – v paths connecting any two distinct vertices u and v. And the rainbow k-connectivity rck (G) of G to be the minimum integer l such that there exists a l-edge-coloring which is a rainbow k-connection coloring. In [4], the authors determined the rck (G) of the complete graph Kn and r-regular complete bipartite graphs Kr, r . In this paper, we study the rainbow k-connectivity of the Cartesian product graphs K 2□Kn for some k. We determine rck (K 2□Kn ) for k = 2, 3, 4 and prove that there exists Cartesian product graphs K 2□Kn such that rck (K 2□Kn ) = 3 for each integer k ≥ 2.

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.

The rainbow connection number of a watermill graph

Let G be a simple, finite and undirected connected graph. An edge-colored graph G is called rainbow connected, if any two vertices in the graph are connected by a path which the edges have distinct colors. Such a path is called rainbow path. An edge-coloring on a graph G is a map . The smallest number k of colors needed in order to make G rainbow connected is called the rainbow connection number of G, denoted by rc(G). The concepts of rainbow connection were introduced by Chartrand et. al. in 2008. Since then many classes of graphs are studied to find its rainbow connection number. The corona product is called a sun graph with 2n vertices. Let Pm be a path with m vertices. The watermill graph is a Cartesian product of graphs and Pm, which is denoted by W M(m, n). In this paper, we determine the rainbow connection number of a watermill graph W M(m, n).

The 2nd International Conference of Combinatorics, Graph Theory, and Network Topology

The Second International Conference on Combinatorics, Graph Theory, and Network Topology 2018DafikEditor in Chief of International Conference on Combinatorics, Graph Theory, and Network Topology 2018E-mail: d.dafik@unej.ac.idWe would like to express our gratitude to all participant who were joining “The Second International Conference on Combinatorics, Graph Theory, and Network Topology (ICCGANT)”. It is the 2nd International conference held by the CGANT Research Group held by University of Jember in cooperation with Indonesian Combinatorics Society (INACOBMS) on 24-25 November 2018. The conference is held to welcome participants from many countries, with broad and diverse research interests of mathematics especially combinatorical study. The mission is to become an annual international forum in the future, where, civil society organization and representative, research students, academics and researchers, scholars, scientist, teachers and practitioners from all over the world could meet in and exchange an idea to share and to discuss theoretical and practical knowledge about mathematics and its applications. The aim of the second conference is to present and discuss the latest research that contributes to the sharing of new theoretical, methodological and empirical knowledge and a better understanding in the area mathematics, application of mathematics as well as mathematics education. The themes of this conference are as follows: (1) Connection of distance to other graph properties, (2) Degree/diameter problem, (3) Distance-transitive and distance-regular graphs, (4) Metric dimension and related parameters, (5) Cages and eccentric graphs, (6) Cycles and factors in graphs, (7) Large graphs and digraphs, (8) Spectral Techniques in graph theory, (9) Ramsey numbers, (10) Dimensions of graphs, (11) Communication networks, (12) Coding theory, (13) Cryptography, (14) Rainbow connection, (15) Graph labelings and coloring, (16) Applications of graph theory. The topics are not limited to the above themes but they also include the mathematical application research of interest in general including mathematics education, such as: (1) Applied Mathematics and Modelling, (2) Applied Physics: Mathematical Physics, Biological Physics, Chemistry Physics, (3) Applied Engineering: Mathematical Engineering, Mechanical engineering, Informatics Engineering, Civil Engineering, (4) Statistics and Its Application, (5) Pure Mathematics (Analysis, Algebra and Geometry), (6) Mathematics Education, (7) Literacy of Mathematics, (8) The Use of ICT Based Media In Mathematics Teaching and Learning, (9) Technological, Pedagogical, Content Knowledge for Teaching Mathematics, (10) Students Higher Order Thinking Skill of Mathematics, (11) Contextual Teaching and Realistic Mathematics, (12) Science, Technology, Engineering, and Mathematics Approach, (13) Local Wisdom Based Education: Ethnomathematics, (14) Showcase of Teaching and Learning of Mathematics, (16) The 21st Century Skills: The Integration of 4C Skill in Teaching Math.The participants of this ICCGANT 2018 were 375 participants consisting research students, academics and researchers, scholars, scientist, teachers and practitioners from many countries. The selected papers to be published on IOP Conference Series: Journal of Physics are 110 papers.On behalf of the organizing committee, finally we gratefully acknowledge the support from the University of Jember of this conference. We would also like to extend our thanks to all lovely participants who have been joining this unforgettable and valuable event..Prof. Dafik, M.Sc., Ph.D.List of The Committees available in this PDF.

The analysis of student’s combinatorial thinking skill based on their cognitive style under the implementation of research based learning in the total rainbow connection study

Combinatorial thinking skill is one of the skills that someone should have. Combinatorial thinking process provides the systematic steps in solving a problem. The purpose of this research was to analyze the level of the students’ combinatorial thinking based on their cognitive styles. This research used a mixed method that combined qualitative method and quantitative method. The research subject consist of two classes, namely control class with 41 students and experimental class with 43 students. The data collection techniques in this research used GEFT (Group Embedded Figures Test), combinatorial thinking tests (pretest and posttest), and interview. After solving the problem, we analyzed the students’ level of combinatorial thinking skill, namely low, good, and high while the cognitive style we categorize into Field Dependent and Field Independent and we applied research based learning to improve the level of combinatorial thinking skill. The research results showed that based on the results of the implementation of research based learning, 12% of students were categorized in the level of low combinatorial thinking skill, 53% were categorized in the level of good combinatorial thinking skill, and 35% in the level of high combinatorial thinking skill. The inferential statistical results of the independent sample t-test on the posttest results showed that the sig (2-tailed) value was 0.006 (p ≤ 0.05) so that it was significant. This showed that there was a significant difference between the control class and the experimental class from the results of students’ test after the implementation of research based learning. It can be concluded that the application of research based learning can improve the students’ combinatorial thinking skills based on their cognitive styles in solving the total rainbow connection problem.

Rainbow Connectivity Using a Rank Genetic Algorithm: Moore Cages with Girth Six

Arainbowt-coloringof at-connected graphGis an edge coloring such that for any two distinct verticesuandvofGthere are at leasttinternally vertex-disjoint rainbow(u,v)-paths. In this work, we apply a Rank Genetic Algorithm to search for rainbowt-colorings of the family of Moore cages with girth six(t;6)-cages. We found that an upper bound in the number of colors needed to produce a rainbow 4-coloring of a(4;6)-cage is 7, improving the one currently known, which is 13. The computation of the minimum number of colors of a rainbow coloring is known to be NP-Hard and the Rank Genetic Algorithm showed good behavior finding rainbowt-colorings with a small number of colors.

The total rainbow connection on comb product of cycle and path graphs

All graphs are finite, connected, simple, non-trivial, and undirected. The graph in this study with V(G) as vertex set and E(G) as edge set. Total-colored graph G is a total connected rainbow of any two vertices which are connected by, at least, one total rainbow path which is the edges and internal vertices which have distinct colors. The total rainbow coloring is a natural extension of edge rainbow coloring and vertex rainbow coloring. 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. This paper shows the lower and upper bounds of trc(G) and the result of the total rainbow connection number of comb product of cycle and path graphs.

Rainbow connection number of generalized composition

Let G be a connected graph with V(G)={v1,…,vn}. The rainbow connection number rc(G) is the smallest k for which there is a map γ:E(G)→{1,…,k} such that any two vertices can be connected by a path whose edge colors are all distinct. The generalized composition G[H1,…,Hn] is obtained by replacing each vi with the graph Hi. We prove rc(G[H1,…,Hn])=diam(G) if each Hi has at least diam(G)≥4 vertices, improving known upper bounds of Basavaraju et al. and Gologranc et al. (2014). We prove the same result when diam(G)=3 but with some additional conditions. When diam(G)=2, we show that the largest possible value of rc(G[H1,…,Hn]) is related to whether every vertex of G is contained in a triangle or not.

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.

Odd connection and odd vertex-connection of graphs

In the last years, connection concepts such as rainbow connection and proper connection appeared in graph theory and received a lot of attention. In this paper, we present a general concept of connection in graphs. As a particular case, we introduce the odd connection number and the odd vertex-connection number of a graph. Furthermore, we compute and study the odd connection number and the odd vertex-connection number of graphs of various graph classes.

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.

Rainbow Connection Number and Chromatic Index of Rough Ideal based Rough Edge Cayley Graph

Rainbow connection number and chromatic index are two significant parameters in the study ofgraph theory. In this work, rainbow connection number and chromatic index of Rough Ideal based Rough Edge Cayley Graph G(T(J)) are evaluated. We prove that the rainbow connection number of G(T(J)) is 2 and the chromatic index of G(T(J)) is 2(2n^m)(3m^1):Rainbow connection number and chromatic index are two significant parameters in the study of graph theory. In this work, rainbow connection number and chromatic index of Rough Ideal based Rough Edge Cayley Graph are evaluated. We prove that the rainbow connection number of is 2 and the chromatic index of is .

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.

Rainbow connectivity of Moore cages of girth 6

Let G be an edge-colored graph. A path P of G is said to be rainbow if no two edges of P have the same color. An edge-coloring of G is a rainbowt-coloring if for any two distinct vertices u and v of G there are at least t internally vertex-disjoint rainbow (u,v)-paths. The rainbowt-connectivityrct(G) of a graph G is the minimum integer j such that there exists a rainbow t-coloring using j colors. A (k;g)-cage is a k-regular graph of girth g and minimum number of vertices denoted n(k;g). In this paper we focus on g=6. It is known that n(k;6)≥2(k2−k+1) and when n(k;6)=2(k2−k+1) the (k;6)-cage is called a Moore cage. In this paper we prove that the rainbow k-connectivity of a Moore (k;6)-cage G satisfies that k≤rck(G)≤k2−k+1. It is also proved that the rainbow 3-connectivity of the Heawood graph is 6 or 7.

Rainbow colouring of Oxide Networks

A path in an edge coloured graph with no two edges sharing the same colour is called a rainbow path. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $k$ for which there exists an $k-$ edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. It is known that computing the rainbow connection number of a graph is $NP-$Hard. So, it is interesting to compute $rc(G)$ for any given graph $G$. In this paper, we compute the rainbow connection number for oxide networks.