Rainbow Tree Research Articles

Complete bipartite graphs without small rainbow subgraphs

Motivated by bipartite Gallai–Ramsey type problems, we consider edge-colorings of complete bipartite graphs without rainbow tree and matching. Given two graphs G and H, and a positive integer k, define the bipartite Gallai–Ramsey numberbgrk(G:H) as the minimum number of vertices n such that n2≥k and for every N≥n, any coloring (using all k colors) of the complete bipartite graph KN,N contains a rainbow copy of G or a monochromatic copy of H. In this paper, we first describe the structures of a complete bipartite graph Kn,n without rainbow P4+ and 3K2, respectively, where P4+ is the graph consisting of a P4 with one extra edge incident with an interior vertex. Furthermore, we determine the exact values or upper and lower bounds on bgrk(G:H) when G is a 3-matching or a 4-path or P4+, and H is a bipartite graph.

Finding Large Rainbow Trees in Colourings of $K_{n,n}$

A subgraph of an edge-coloured graph is called rainbow if all of its edges have distinct colours. An edge-colouring is called locally $k$-bounded if each vertex is incident with at most $k$ edges of the same colour. Recently, Montgomery, Pokrovskiy and Sudakov showed that for large $n$, a certain locally 2-bounded edge-colouring of the complete graph $K_{2n+1}$ contains a rainbow copy of any tree with $n$ edges, thereby resolving a long-standing conjecture by Ringel: For large $n$, $K_{2n+1}$ can be decomposed into copies of any tree with $n$ edges. In this paper, we employ their methods to show that any locally $k$-bounded edge-colouring of the complete bipartite graph $K_{n,n}$ contains a rainbow copy of any tree $T$ with $(1- o(1))n/k$ edges. We show that this implies that every tree with $n$ edges packs at least $n$ times into $K_{n+o(1),n+o(1)}$. We conjecture that for large $n$, $K_{n,n}$ can be decomposed into $n$ copies of any tree with $n$ edges.

교실 공간으로서의 벽면에서 이루어지는 유아 참여적 전시의 의미

Objectives The purpose of this study is to find out the process and educational meaning of young children’s participatory displays on the wall as a classroom space.
 Methods To this end, from the beginning of September 2019 to the end of February 2020, the 4-year-old class was observed twice a week, and a teacher and children were interviewed, and collection of artifacts were conducted. The collected data was analyzed using the thematic analysis method, taking the content of the participatory wall display for young children suggested by Kang, Jinju and Goh, Eun Kyoung (2022) as an interpretive framework.
 Results As a result of this study, the tree display, ‘rainbow tree’ named by children and their parents was put in seven different shapes on the wall of the class for 6 months. The process and meaning of the wall display: First, as breaking the space boundary, the rainbow tree changed its appearance in various ways, and a wall display changed the space of our class. Second, as breaking the time boundary, the current interest of our class was revealed, and a wall display was made in which our stories with serendipity were continuously made. Fourth, as breaking the boundaries of authority, the children's participation became more diverse and the wall displays were made to tell our authentic stories.
 Conclusions It was discussed that the young children’s participatory wall display constituted the narrative of young children and was a child-centered educational experience in which children could actively participate in learning, which the wall can be a meaningful space within the classroom as a place for playing and documenting at the same time.

Rainbow trees in uniformly edge‐colored graphs

AbstractWe obtain sufficient conditions for the emergence of spanning and almost‐spanning bounded‐degree rainbow trees in various host graphs, having their edges colored independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform coloring of , using a palette of size , a.a.s. admits a rainbow copy of any given bounded‐degree tree on at most vertices, where is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon et al. pertaining to the embedding of bounded‐degree almost‐spanning prescribed trees in , where is independent of . Given an ‐vertex graph with minimum degree at least , where is fixed, we use our aforementioned result in order to prove that a uniform coloring of the randomly perturbed graph , using colors, where is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded‐degree spanning tree. This can be viewed as a rainbow variant of a result by Krivelevich et al. who proved that , where is independent of , a.a.s. admits a copy of any given bounded‐degree spanning tree. Finally, and with as above, we prove that a uniform coloring of using colors a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.

The strong 3-rainbow index of edge-comb product of a path and a connected graph

Let G be a connected and edge-colored graph of order n , where adjacent edges may be colored the same. A tree in G is a rainbow tree if all of its edges have distinct colors. Let k be an integer with 2 ≤ k ≤ n . The minimum number of colors needed in an edge coloring of G such that there exists a rainbow tree connecting S with minimum size for every k -subset S of V ( G ) is called the strong k -rainbow index of G , denoted by s r x k ( G ) . In this paper, we study the s r x 3 of edge-comb product of a path and a connected graph, denoted by P n o ⊳ e H . It is clearly that | E ( P n o ⊳ e H )| is the trivial upper bound for s r x 3 ( P n o ⊳ e H ) . Therefore, in this paper, we first characterize connected graphs H with s r x 3 ( P n o ⊳ e H )=| E ( P n o ⊳ e H )| , then provide a sharp upper bound for s r x 3 ( P n o ⊳ e H ) where s r x 3 ( P n o ⊳ e H )≠| E ( P n o ⊳ e H )| . We also provide the exact value of s r x 3 ( P n o ⊳ e H ) for some connected graphs H .

The strong 3-rainbow index of some certain graphs and its amalgamation

We introduce a strong \(k\)-rainbow index of graphs as modification of well-known \(k\)-rainbow index of graphs. A tree in an edge-colored connected graph \(G\), where adjacent edge may be colored the same, is a rainbow tree if all of its edges have distinct colors. Let \(k\) be an integer with \(2\leq k\leq n\). The strong \(k\)-rainbow index of \(G\), denoted by \(srx_k(G)\), is the minimum number of colors needed in an edge-coloring of \(G\) so that every \(k\) vertices of \(G\) is connected by a rainbow tree with minimum size. We focus on \(k=3\). We determine the strong \(3\)-rainbow index of some certain graphs. We also provide a sharp upper bound for the strong \(3\)-rainbow index of amalgamation of graphs. Additionally, we determine the exact values of the strong \(3\)-rainbow index of amalgamation of some graphs.

Embedding rainbow trees with applications to graph labelling and decomposition

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares. Since then rainbow structures have been the focus of extensive research and have found applications in the areas of graph labelling and decomposition. An edge-colouring is locally k -bounded if each vertex is contained in at most k edges of the same colour. In this paper we prove that any such edge-colouring of the complete graph K_n contains a rainbow copy of every tree with at most (1-o(1))n/k vertices. As a locally k -bounded edge-colouring of K_n may have only (n-1)/k distinct colours, this is essentially tight. As a corollary of this result we obtain asymptotic versions of two long-standing conjectures in graph theory. Firstly, we prove an asymptotic version of Ringel’s conjecture from 1963, showing that any n -edge tree packs into the complete graph K_{2n+o(n)} to cover all but o(n^2) of its edges. Secondly, we show that all trees have an almost-harmonious labelling. The existence of such a labelling was conjectured by Graham and Sloane in 1980. We also discuss some additional applications.

The strong 3-rainbow index of edge-amalgamation of some graphs

Let G be a nontrivial, connected, and edge-colored graph of order n ≥ 3, where adjacent edges may be colored the same. Let k be an integer with 2 ≤ k ≤ n. A tree T in G is a rainbow tree if no two edges of T are colored the same. For S ⊆ V G , the Steiner distance d S of S is the minimum size of a tree in G containing S . An edge-coloring of G is called a strong k -rainbow coloring if for every set S of k vertices of G there exists a rainbow tree of size d S in G containing S . The minimum number of colors needed in a strong k -rainbow coloring of G is called the strong k -rainbow index srxk G of G. In this paper, we study the strong 3-rainbow index of edge-amalgamation of graphs. We provide a sharp upper bound for the srx3 of edge-amalgamation of graphs. We also determine the srx3 of edge-amalgamation of some graphs.

Complete graphs with no rainbow tree

AbstractWe study colorings of the edges of the complete graph . For some graph , we say that a coloring contains a rainbow , if there is an embedding of into , such that all edges of the embedded copy have pairwise distinct colors. The main emphasis of this paper is a classification of those forbidden rainbow graphs that force a low number of vertices of a high color degree, along with some specifications and more general information in certain cases. Those graphs turn out to be of two special types of trees of small diameter.

A new approach for the rainbow spanning forest problem

Given an edge-colored graph G, a tree with all its edges with different colors is called a rainbow tree. The rainbow spanning forest (RSF) problem consists of finding a spanning forest of G, with the minimum number of rainbow trees. In this paper, we present an integer linear programming model for the RSF problem that improves a previous formulation for this problem. A GRASP metaheuristic is also implemented for providing fast primal bounds for the exact method. Computational experiments carried out over a set of random instances show the effectiveness of the strategies adopted in this work, solving problems in graphs with up to 100 vertices.

Decompositions into spanning rainbow structures

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares and has been the focus of extensive research ever since. Euler posed a problem equivalent to finding properly $n$-edge-coloured complete bipartite graphs $K_{n,n}$ which can be decomposed into rainbow perfect matchings. While there are proper edge-colourings of $K_{n,n}$ without even a single rainbow perfect matching, the theme of this paper is to show that with some very weak additional constraints one can find many disjoint rainbow perfect matchings. In particular, we prove that if some fraction of the colour classes have at most $(1-o(1)) n$ edges then one can nearly-decompose the edges of $K_{n,n}$ into edge-disjoint perfect rainbow matchings. As an application of this, we establish in a very strong form a conjecture of Akbari and Alipour and asymptotically prove a conjecture of Barat and Nagy. Both these conjectures concern rainbow perfect matchings in edge-colourings of $K_{n,n}$ with quadratically many colours. Using our techniques, we also prove a number of results on near-decompositions of graphs into other rainbow structures like Hamiltonian cycles and spanning trees. Most notably, we prove that any properly coloured complete graph can be nearly-decomposed into spanning rainbow trees. This asymptotically proves the Brualdi-Hollingsworth and Kaneko-Kano-Suzuki conjectures which predict that a perfect decomposition should exist under the same assumptions.

The 3-rainbow index of some graphs that constructed by joining a graph with a trivial graph

Let G = (V, E) be a nontrivial, finite, and connected graph. A tree T in an edge-colored graph is said to be rainbow tree, if no two edges on the tree have the same color. An edge-coloring of G is called 3-rainbow coloring, if for any three vertices in G there exists a rainbow tree connecting them. The 3-rainbow index of G, denoted by rx3(G), is the minimum number of colors that are needed in a 3-rainbow coloring of G. The join of G1 and G2, denoted by G = G1 + G2, is the graph with vertex set V(G1) ∪ V(G2) and edge set E(G1) ∪ E(G2) ∪ {uv|u ∊ V(G1),v ∊ V(G2)}. In this paper, we determine rx3(G) where G is the join of a graph with a trivial graph.

The 3-rainbow index of amalgamation of some graphs with diameter 2

Let G = (V, E) be a nontrivial, connected, and edge-colored graph with n vertices, and let k be an integer with 2 ≤ k ≤ n. A tree T in G is a rainbow tree, if no two edges of T receive the same color. A k-rainbow coloring of G is an edge coloring of G having property that for every set S of k vertices of G, there exists a rainbow tree T such that S ⊆ V (T). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by rxk(G). The distance d(x,y) of two vertices x and y in G is the length of a shortest x — y path in G. The greatest distance between any two vertices in G is the diameter of G, denoted by diam(G). Let {G1, G2,…, Gt} be a finite collection of graphs and each graph Gi have a fixed vertex v0i called a terminal. The amalgamation of G1, G2,…, Gt, denoted by Amal(G1, v0l, G2, v02 …, Gt, v0t), is a graph obtained by taking all the s and identifying their terminals. In case Gi ≅ G and v0i = u, the amalgamation of G1, G2,…, Gt is denoted by Amal(G,t,u). In this paper, we determine the 3-rainbow index of amalgamation of some graphs Amal(G,t,u) with diam(G) = 2.

On the number of rainbow spanning trees in edge-colored complete graphs

A spanning tree of a properly edge-colored complete graph, Kn, is rainbow provided that each of its edges receives a distinct color. In 1996, Brualdi and Hollingsworth conjectured that if K2m is properly (2m−1)-edge-colored, then the edges of K2m can be partitioned into m rainbow spanning trees except when m=2. By means of an explicit, constructive approach, in this paper we construct ⌊6m+9∕3⌋ mutually edge-disjoint rainbow spanning trees for any positive value of m. Not only are the rainbow trees produced, but also some structure of each rainbow spanning tree is determined in the process. This improves upon best constructive result to date in the literature which produces exactly three rainbow trees.

Linearly many rainbow trees in properly edge-coloured complete graphs

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. In this paper we discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi–Hollingsworth Conjecture, Constantine's Conjecture, and the Kaneko–Kano–Suzuki Conjecture. We show that in every proper edge-colouring of Kn there are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method we also show that every properly (n−1)-edge-coloured Kn has n/9−6 edge-disjoint rainbow trees, giving further improvement on the Brualdi–Hollingsworth Conjecture.

Edge-disjoint rainbow trees in properly coloured complete graphs

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. We discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine's Conjecture, and the Kaneko-Kano-Suzuki Conjecture. The main result which we discuss is that in every proper edge-colouring of Kn there are 10−6n edge-disjoint isomorphic spanning rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method it is also possible to show that every properly (n−1)-edge-coloured Kn has n/9 edge-disjoint spanning rainbow trees, giving a further improvement on the Brualdi-Hollingsworth Conjecture.

3-Rainbow Index and Forbidden Subgraphs

A tree in an edge-colored connected graph G is called a rainbow tree if no two edges of it are assigned the same color. For a vertex subset $$S\subseteq V(G)$$ , a tree is called an S-tree if it connects S in G. A k-rainbow coloring of G is an edge-coloring of G having the property that for every set S of k vertices of G, there exists a rainbow S-tree in G. The minimum number of colors that are needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by $${\mathrm {rx}}_k(G)$$ . The Steiner distance d(S) of a set S of vertices of G is the minimum size of an S-tree T. The k-Steiner diameter $${\mathrm {sdiam}}_k(G)$$ of G is defined as the maximum Steiner distance of S among all sets S with k vertices of G. In this paper, we focus on the 3-rainbow index of graphs and find all finite families $$\mathcal {F}$$ of connected graphs, for which there is a constant $$C_\mathcal {F}$$ such that, for every connected $$\mathcal {F}$$ -free graph G, $${\mathrm {rx}}_3(G)\le {\mathrm {sdiam}}_3(G)+C_\mathcal {F}$$ .

The $$(k,\ell )$$ ( k , ℓ ) -Rainbow Index for Complete Bipartite and Multipartite Graphs

A tree in an edge-colored graph G is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers k, \(\ell \) with \(k\ge 3\), the \((k,\ell )\)-rainbow index\(rx_{k,\ell }(G)\) of G is the minimum number of colors needed in an edge-coloring of G such that for any set S of k vertices of G, there exist \(\ell \) internally disjoint rainbow trees connecting S. This concept was introduced by Chartrand et al. in 2010. It is very difficult to determine the \((k,\ell )\)-rainbow index for a general graph. Chartrand et al. determined the (k, 1)-rainbow index of all unicyclic graphs and the \((3,\ell )\)-rainbow index of complete graphs for \(\ell =1,2\). We showed that for every pair of positive integers \(k,\ell \) with \(k\ge 3\), there exists a positive integer \(N=N(k,\ell )\) such that \(rx_{k,\ell }(K_{n})=k\) for every integer \(n\ge N\), which settled down a conjecture of Chartrand et al. In this paper, we use probabilistic method and bipartite Ramsey numbers to obtain similar results of the \((k,\ell )\)-rainbow index for complete bipartite graphs. For complete multipartite graphs, we get similar results for most cases, however, since there is no any result on the multipartite Ramsey numbers in general, we can only get a value that differs by 1 from the exact value for some cases.

The $$(k,\ell )$$ ( k , ℓ ) -Rainbow Index of Random Graphs

A tree in an edge-colored graph G is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers k, \(\ell \) with \(k\ge 3\), the \((k,\ell )\)-rainbow index\(rx_{k,\ell }(G)\) of G is the minimum number of colors needed in an edge-coloring of G such that for any set S of k vertices of G, there exist \(\ell \) internally disjoint rainbow trees connecting S. This concept was introduced by Chartrand et. al., and there have been very few known results about it. In this paper, we establish a sharp threshold function for \(rx_{k,\ell }(G_{n,p})\le k\) and \(rx_{k,\ell }(G_{n,M})\le k,\) respectively, where \(G_{n,p}\) and \(G_{n,M}\) are the usually defined random graphs.

Note on the upper bound of the rainbow index of a graph

A path in an edge-colored graph G is a rainbow path if every two edges of it receive distinct colors. The rainbow connection number of a connected graph G, denoted by rc(G), is the minimum number of colors that are needed to color the edges of G such that there is a rainbow path connecting every two vertices of G. Similarly, a tree in G is a rainbow tree if no two edges of it receive the same color. The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbow tree containing all the vertices of S (other vertices may also be included in the tree) for each k-subset S of V(G) is called the k-rainbow index of G, denoted by rxk(G), where k is an integer such that 2≤k≤n. Chakraborty et al. got the following result: For every ϵ>0, a connected graph with minimum degree at least ϵn has bounded rainbow connection number, where the bound depends only on ϵ. Krivelevich and Yuster proved that if G has n vertices and the minimum degree δ(G) then rc(G)<20n/δ(G). This bound was later improved to 3n/(δ(G)+1)+3 by Chandran et al. Since rc(G)=rx2(G), a natural problem arises: for a general k determining the true behavior of rxk(G) as a function of the minimum degree δ(G). In this paper, we give upper bounds of rxk(G) in terms of the minimum degree δ(G) in different ways, namely, via Szemerédi’s Regularity Lemma, connected 2-step dominating sets, connected (k−1)-dominating sets and k-dominating sets of G.