Related papers: A note on rainbow matchings in properly edge-colou…
Let $G$ be a nontrivial connected, edge-colored graph. An edge-cut $S$ of $G$ is called a rainbow cut if no two edges in $S$ are colored with a same color. An edge-coloring of $G$ is a rainbow disconnection coloring if for every two…
A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree $\Delta$ has a strong edge-colouring with at most $4\Delta+4$ colours. We show…
Given an edge-colored graph $G$, we denote the number of colors as $c(G)$, and the number of edges as $e(G)$. An edge-colored graph is rainbow if no two edges share the same color. A proper $mK_3$ is a vertex disjoint union of $m$ rainbow…
Let $G$ be an edge-colored connected graph. A path of $G$ is called rainbow if its every edge is colored by a distinct color. $G$ is called rainbow connected if there exists a rainbow path between every two vertices of $G$. The minimum…
An edge-cut $R$ of an edge-colored connected graph is called a rainbow-cut if no two edges in the edge-cut are colored the same. An edge-colored graph is rainbow disconnected if for any two distinct vertices $u$ and $v$ of the graph, there…
Aharoni and Berger conjectured that in every proper edge-colouring of a bipartite multigraph by $n$ colours with at least $n+1$ edges of each colour there is a rainbow matching using every colour. This conjecture generalizes a longstanding…
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 edge-colored graph on $n$ vertices. The minimum color degree of $G$, denoted by $\delta^c(G)$, is defined as the minimum number of colors assigned to the edges incident to a vertex in $G$. In 2013, H. Li proved that an…
An edge-coloured path is \emph{rainbow} if its edges have distinct colours. An edge-coloured connected graph is said to be \emph{rainbow connected} if any two vertices are connected by a rainbow path, and \emph{strongly rainbow connected}…
Let $G = (V,E)$ be an $n$-vertex graph and let $c: E \to \mathbb{N}$ be a coloring of its edges. Let $d^c(v)$ be the number of distinct colors on the edges at $v \in V$ and let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$. H. Li proved…
In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge colorings of complete graphs, and we prove that in every proper edge coloring of…
In an edge-colored graph $G$, a rainbow clique $K_k$ is a $k$-complete subgraph in which all the edges have distinct colors. Let $e(G)$ and $c(G)$ be the number of edges and colors in $G$, respectively. In this paper, we show that for any…
A path in an edge-colored graph is called {\em rainbow} if no two edges of it are colored the same. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k\leq \ell$, the {\em rainbow $k$-connection number} $rc_k(G)$ of $G$ is…
An edge-colouring of a graph $G$ can fail to be rainbow for two reasons: either it contains a monochromatic cherry (a pair of incident edges), or a monochromatic matching of size two. A colouring is a proper colouring if it forbids the…
Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v \in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We prove that the same hypothesis…
We consider the problem of finding a large rainbow matching in a random graph with randomly colored edges. In particular we analyze the performance of two greedy algorithms for this problem. The algorithms we study are colored versions of…
The 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. We show…
We consider edge colorings of graphs. An edge coloring is a majority coloring if for every vertex at most half of the edges incident with it are in one color. And edge coloring is a distinguishing coloring if for every non-trivial…
An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that…
A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if there is a rainbow path connecting any two…