Related papers: Short rainbow cycles in edge-colored graphs
Given a family $\mathcal G$ of graphs on a common vertex set $X$, we say that $\mathcal G$ is rainbow connected if for every vertex pair $u,v \in X$, there exists a path from $u$ to $v$ that uses at most one edge from each graph in…
Given a digraph $D$, we denote by $\vec{\alpha}(D)$ the maximum size of an acyclic set of $D$ (i.e. a set of vertices which induces a subdigraph with no directed cycles), and by $\vec\chi(D)$ the minimum number of acyclic sets into which…
The star chromatic index of a graph $G$, denoted by $\chi^\prime_s(G)$, is the smallest integer $k$ for which $G$ admits a proper edge coloring with $k$ colors such that every path and cycle of length four is not bicolored. Let $d$ be the…
Albertson conjectured that if graph $G$ has chromatic number $r$, then the crossing number of $G$ is at least that of the complete graph $K_r$. This conjecture in the case $r=5$ is equivalent to the four color theorem. It was verified for…
A particular case of Caccetta-H\"{a}ggkvist conjecture, says that a digraph of order $n$ with minimum out-degree at least $1/3n$ contains a directed cycle of length at most 3. Recently, Kral, Hladky and Norine proved that a digraph of order…
An edge-colored graph $G$ is rainbow connected, if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are…
One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdos-Renyi random graph G_{n,p} is around p ~ (log n + log log n) / n. Much research has been done to extend this to…
An {\em acyclic edge coloring} of a graph $G$ is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The {\em acyclic chromatic index}…
A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We…
A star edge coloring of a graph $G$ is a proper edge coloring of $G$ without bichromatic paths or cycles of length four. The it star chromatic index, $\chi_{st}^{'} (G ),$ of $G$ is the minimum number $k$ for which $G$ has a star edge…
Let $ n \in \mathbb{N} $ with $ n \geq 3 $, and let $\mathcal{G} = \{G_i:i\in [n]\} $ be a family of $ n $-vertex graphs on a common vertex set $V$, where the graphs in the family do not need to be distinct. A graph $H$ with vertex set $V$…
A star edge-coloring of a graph $G$ is a proper edge-coloring without bichromatic paths or cycles of length four. The smallest integer $k$ such that $G$ admits a star edge-coloring with $k$ colors is the star chromatic index of $G$. In the…
An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and…
A spanning tree of an edge-colored graph is rainbow provided that each of its edges receives a distinct color. In this paper we consider the natural extremal problem of maximizing and minimizing the number of rainbow spanning trees in a…
A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which…
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…
Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored the same. An edge-colored graph $G$ is rainbow disconnected if for every two vertices $u$ and $v$, there…
An edge-colored multigraph $G$ is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color. In the context of multilayered networks we introduce the notion of…
A path in an edge-colored graph is called a proper path if no two adjacent edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…
A path in an edge-colored graph $G$ is rainbow if no two edges of it are colored the same. The graph $G$ is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair…