Rainbow Connection Research Articles

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.

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}.

On the total rainbow connection of the wheel related graphs

Let G = (V(G), E(G)) be a nontrivial connected graph with an edge coloring c : E(G) → {1, 2, …, l}, l ∊ N, with the condition that the adjacent edges may be colored by the same colors. A path P in G is called rainbow path if no two edges of P are colored the same. The smallest number of colors that are needed to make G rainbow edge-connected is called the rainbow edge-connection of G, denoted by rc(G). A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The smallest number of colors that are needed to make G rainbow vertex-connected is called the rainbow vertex-connection of G, denoted by rvc(G). A total-colored path is total-rainbow if edges and internal vertices have distinct colours. The minimum number of colour required to color the edges and vertices of G is called the total rainbow connection number of G, denoted by trc(G). In this paper, we determine the total rainbow connection number of some wheel related graphs such as gear graph, antiweb-gear graph, infinite class of convex polytopes, sunflower graph, and closed-sunflower graph.

Rainbow connections in digraphs

An arc-coloured path in a digraph is rainbow if its arcs have distinct colours. A vertex-coloured path is vertex rainbow if its internal vertices have distinct colours. A totally-coloured path is total rainbow if its arcs and internal vertices have distinct colours. An arc-coloured (resp. vertex-coloured, totally-coloured) digraph D is rainbow connected (resp. rainbow vertex-connected, total rainbow connected) if any two vertices of D are connected by a rainbow (resp. vertex rainbow, total rainbow) path. The rainbow connection number (resp. rainbow vertex-connection number, total rainbow connection number) of a digraph D is the smallest number of colours needed to make D rainbow connected (resp. rainbow vertex-connected, total rainbow connected).In this paper, we study the rainbow connection, rainbow vertex-connection and total rainbow connection numbers of digraphs. We give some properties of these parameters and establish relations between them. The rainbow connection number and the rainbow vertex-connection number of a digraph D are both upper bounded by the order of D, while its total rainbow connection number is upper bounded by twice of its order. In particular, we prove that a digraph of order n has rainbow connection number n if and only if it is Hamiltonian and has three vertices with eccentricity n−1, that it has rainbow vertex-connection number n if and only if it has a Hamiltonian cycle C and three vertices with eccentricity n−1 such that no two of them are consecutive on C, and that it has total rainbow connection number 2n if and only if it has rainbow vertex-connection number n.

The Rainbow Connection: How Music Classrooms Create Safe Spaces for Sexual-Minority Young People

LGBTQ students in many parts of the United States often experience hostility on the part of other students, teachers, and administrators. This article reviews current terminology, examines present-day attitudes and recent literature, and offers suggestions to educators who want to create safe spaces for all students in their classrooms.

Experiences of and barriers to transition-related healthcare among Korean transgender adults: focus on gender identity disorder diagnosis, hormone therapy, and sex reassignment surgery.

OBJECTIVESTransgender people may encounter barriers to transition-related healthcare services. This study aimed to investigate the experiences of transition-related healthcare and barriers to those procedures among transgender adults in Korea.METHODSIn 2017, we conducted a nationwide cross-sectional survey of 278 transgender adults, which named Rainbow Connection Project II, in Korea. We assessed the prevalence of transition-related healthcare, including gender identity disorder (GID) diagnosis, hormone therapy, and sex reassignment surgery. To understand the barriers to those procedures, we also asked participants for their reasons for not receiving each procedure. Further, this study examined their experiences of and the reasons for using non-prescribed hormone medications.RESULTSOf transgender people participated in the survey, 91.0% (n=253/278) were diagnosed with GID, 88.0% (n=243/276) received hormone therapy, and 42.4% (n=115/271) have had any kind of sex reassignment surgery. Cost was the most common barrier to transition-related healthcare among Korean transgender adults. Other common barriers were identified as follows: negative experiences in healthcare settings, lack of specialized healthcare professionals and facilities, and social stigma against transgender people. Among those who had taken hormone medications, 25.1% (n=61/243) reported that they had ever purchased them without a prescription.CONCLUSIONSOur findings suggest that barriers to transition-related healthcare exist in Korea and constrain transgender individuals’ safe access to the needed healthcare. Institutional interventions are strongly recommended to improve access to transition-related healthcare. These interventions include provision of programs to train Korean healthcare professionals and expansion of national health insurance to include these procedures.

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.

Generalized rainbow connection of graphs and their complements

Generalized rainbow connection of graphs and their complements

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.

Health disparities between lesbian, gay, and bisexual adults and the general population in South Korea: Rainbow Connection Project I.

OBJECTIVESThis study aims to investigate health disparities between lesbian, gay, and bisexual (LGB) adults and the general population in Korea, where there is low public acceptance of sexual minorities and a lack of research on the health of sexual minorities.METHODSThe research team conducted a nationwide survey of 2,335 Korean LGB adults in 2016. Using the dataset, we estimated the age-standardized prevalence ratios (SPRs) for poor self-rated health, musculoskeletal pain, depressive symptoms, suicidal behaviors, smoking, and hazardous drinking. We then compared the SPRs of the LGB adults and the general population which participated in three different nationally representative surveys in Korea. SPRs were estimated for each of the four groups (i.e., gay men, bisexual men, lesbians, and bisexual women).RESULTSKorean LGB adults exhibited a statistically significantly higher prevalence of depressive symptoms, suicidal ideation and attempts, and musculoskeletal pain than the general population. Lesbian and bisexual women had a higher risk of poor self-rated health and smoking than the general women population, whereas gay and bisexual men showed no differences with the general men population. Higher prevalence of hazardous drinking was observed among lesbians, gay men, and bisexual women compared to the general population, but was not observed in bisexual men.CONCLUSIONSThe findings suggest that LGB adults have poorer health conditions compared to the general population in Korea. These results suggest that interventions are needed to address the health disparities of Korean LGB adults.

A Note on the Rainbow Connection of Random Regular Graphs

We prove that a random 3-regular graph has rainbow connection number $O(\log n)$. This completes the remaining open case from Rainbow connection of random regular graphs, by Dudek, Frieze and Tsourakakis.

The Analysis of Student’s Creative Thinking Skills in Solving "Rainbow Connection" Problem through Research Based Learning

Creative thinking skills are needed in the 21st century learning. According to the P21 platform (Partnership for 21stcentury learning), someone will survive in the 21st century if they have some skills of one of them is creative thinking skill. Byapplying Research Based Learning (RBL), 64 students were given a problem namely Rainbow Connection. Through qualitativeresearch, student results are analyzed to know the level of their creative thinking skills. The results shows that the data ofstudents's creative thinking level are as follows: (i) In class A, 28 students have creative thinking skills of level 4, 2 students havecreative thinking skills of level 3, and 4 students have creative thinking skills of level 2. (ii) In class B, 20 students have creativethinking skills of level 4, 2 students have creative thinking skills of level 3, and 8 students have creative thinking skills of level 2.It can be concluded that the level of students creative thinking skills in solving Rainbow Connection problems through RBL arerelatively high.

Rainbow vertex connection of digraphs

An edge-coloured path is rainbow if its edges have distinct colours. An edge-coloured connected graph is said to be rainbow connected if any two vertices are connected by a rainbow path, and strongly rainbow connected if any two vertices are connected by a rainbow geodesic. The (strong) rainbow connection number of a connected graph is the minimum number of colours needed to make the graph (strongly) rainbow connected. These two graph parameters were introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). As an extension, Krivelevich and Yuster proposed the concept of rainbow vertex-connection. The topic of rainbow connection in graphs drew much attention and various similar parameters were introduced, mostly dealing with undirected graphs. Dorbec, Schiermeyer, Sidorowicz and Sopena extended the concept of the rainbow connection to digraphs. In this paper, we consider the (strong) rainbow vertex-connection number of digraphs. Results on the (strong) rainbow vertex-connection number of biorientations of graphs, cycle digraphs, circulant digraphs and tournaments are presented.

Rainbow connection number and graph operations

A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. An edge-colored graph G is rainbow connected if every two distinct vertices are connected by a rainbow path. The rainbow connection numberrc(G) of G is the smallest number of colors that are needed in order to make G rainbow connected. In this paper, we study bounds of rainbow connection number of some graph operations, such as the union of two graphs, adding edges, deleting edges, and adding vertices and edges. Moreover, we also study the following extremal problem. Let k and n be two integers such that 1≤k≤ℓ<n. Find the smallest integer f(n,k,ℓ) such that for each graph G of order n and diameter k, there exists an edge set F⊆E(G¯) satisfying |F|≤f(n,k,ℓ) and rc(G+F)≤ℓ.

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 numbers of Cayley digraphs on abelian groups

A directed path in an edge colored digraph is said to be a rainbow path if no two edges on this path share the same color. An edge colored digraph Γ is rainbow connected if any two distinct vertices can be reachable from each other through rainbow paths. The rc-number of a digraph Γ is the smallest number of colors that are needed in order to make Γ rainbow connected. In this paper, we investigate the rc-numbers of Cayley digraphs on abelian groups and present an upper bound for such digraphs. In addition, we consider the rc-numbers of bi-Cayley graphs on abelian groups.

On total rainbow k-connected graphs

A total-colored graph G is total rainbow connected if any two vertices are connected by a path whose edges and inner vertices have distinct colors. A graph G is total rainbow k-connected if there is a total-coloring of G with k colors such that G is total rainbow connected. The total rainbow connection number, denoted by trc(G), of a graph G is the smallest k to make G total rainbow k-connected. For n, k ≥ 1, define h(n, k) to be the minimum size of a total rainbow k-connected graph G of order n. In this paper, we prove a sharp upper bound for trc(G) in terms of the number of vertex-disjoint cycles of G. We also compute exact values and upper bounds for h(n, k).

An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

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.

Rainbow connection number of graphs with diameter 3

Rainbow connection number of graphs with diameter 3