Related papers: Rainbow trees in uniformly edge-coloured graphs
A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices.…
Let $G_{n,p}^{[\kappa]}$ denote the space of $n$-vertex edge coloured graphs, where each edge occurs independently with probability $p$. The colour of each existing edge is chosen independently and uniformly at random from the set…
An edge-colored graph $G$ is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number, denoted by $rc(G)$, is the minimum number of colors needed to…
Let $G$ be an edge-coloured graph. A rainbow subgraph in $G$ is a subgraph such that its edges have distinct colours. The minimum colour degree $\delta^c(G)$ of $G$ is the smallest number of distinct colours on the edges incident with a…
Let $G = (V, E)$ be a graph on $n$ vertices, and let $c: E \to P$, where $P$ is a set of colors. Let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011,…
Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a coloring of the edges of the complete graph K_n such that at each vertex every color appears only constantly many times. In 1979,…
Let $G$ be an edge-colored graph on $n$ vertices. For a vertex $v$, the \emph{color degree} of $v$ in $G$, denoted by $d^c(v)$, is the number of colors appearing on the edges incident with $v$. Denote by $\delta^c(G)=\min\{d^c(v):v\in…
We call an edge colouring of a graph G a rainbow colouring if every pair of vertices is joined by a rainbow path, i.e., a path where no two edges have the same colour. The minimum number of colours required for a rainbow colouring of the…
A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on…
Let $G$ be a connected multigraph with $n$ vertices, and suppose $G$ has been edge-colored with $n-1$ colors so that each color class induces a spanning tree. Rota's Basis Conjecture for graphic matroids posits that one can find $n-1$…
Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this…
A rainbow matching in an edge-colored graph is a matching in which no two edges have the same color. The color degree of a vertex v is the number of different colors on edges incident to v. Kritschgau [Electron. J. Combin. 27(2020)] studied…
An edge-colored graph $G$, where adjacent edges may be colored the same, is rainbow connected if any two vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of a connected graph…
We study the following rainbow version of subgraph containment problems in a family of (hyper)graphs, which generalizes the classical subgraph containment problems in a single host graph. For a collection $\textbf{G}=\{G_1, G_2,\ldots,…
In this paper we show how to use simple partitioning lemmas in order to embed spanning graphs in a typical member of $G(n,p)$. Let the \emph{maximum density} of a graph $H$ be the maximum average degree of all the subgraphs of $H$. First,…
We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph $\Gamma$ is called $H$-rainbow saturated if $\Gamma$ does not contain a rainbow copy of $H$ and adding an edge of any color to $\Gamma$…
An edge-colored graph $G$, where adjacent edges may be colored the same, is rainbow connected if any two vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of a connected graph…
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. The problem of finding rainbow subgraphs goes back to the work of Euler on transversals in Latin squares and was extensively studied since then.…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. Our main result implies that, given any optimal colouring of a sufficiently large complete graph $K_{2n}$, there exists a decomposition of…
A total-colored graph is a graph $G$ such that both all edges and all vertices of $G$ are colored. A path in a total-colored graph $G$ is a total rainbow path if its edges and internal vertices have distinct colors. A total-colored graph…