Related papers: Rainbow vertex-connection and forbidden subgraphs
A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored the same. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…
Let $G$ be a finite non-abelian group. The non-commuting graph $\Gamma_G$ of $G$ has the vertex set $G\setminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent if $xy\ne yx$, where $Z(G)$ is the center of $G$. We prove that the…
Let $G$ be a connected graph. The notion \emph{the rainbow connection number $rc(G)$} of a graph $G$ was introduced recently by Chartrand et al. Basavaraju et al. showed that for every bridgeless graph $G$ with radius $r$, $rc(G)\leq…
The concept of rainbow disconnection number of graphs was introduced by Chartrand et al. in 2018. Inspired by this concept, we put forward the concepts of rainbow vertex-disconnection and proper disconnection in graphs. In this paper, we…
We study a new variant of graph coloring by adding a connectivity constraint. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph $G$ is said to be…
A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a…
A rainbow graph is a graph that admits a vertex-coloring such that every color appears exactly once in the neighborhood of each vertex. We investigate some properties of rainbow graphs. In particular, we show that there is a bijection…
An edge-colored graph $G$ is called properly colored if no two adjacent edges share a color in $G$. An edge-colored connected graph $G$ is called properly connected if between every pair of distinct vertices, there exists a path that is…
Given a finite family $\mathcal{F}$ of graphs, we say that a graph $G$ is "$\mathcal{F}$-free" if $G$ does not contain any graph in $\mathcal{F}$ as a subgraph. A vertex-colored graph $H$ is called "rainbow" if no two vertices of $H$ have…
The rainbow connection number $rc(G)$ and the rainbow vertex-connection number $rvc(G)$ of a graph $G$ were introduced by Chartrand et al. and Krivelevich and Yuster, respectively. Good upper bounds in terms of minimum degree $\delta$ were…
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…
A {\it rainbow matching} in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function f(\delta) such that a properly edge-colored graph G with minimum degree \delta and order at least…
Let $G$ be an edge-colored graph. We use $e(G)$ and $c(G)$ to denote the number of edges of $G$ and the number of colors appearing on $E(G)$, respectively. For a vertex $v\in V(G)$, the \emph{color neighborhood} of $v$ is defined as the set…
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…
A tree in an edge colored graph 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 $k\geq 3$, the \emph{$(k,\ell)$-rainbow index} $rx_{k,\ell}(G)$ of $G$ is the…
A graph is said to be {\it total-colored} if all the edges and vertices of the graph are colored. A path in a total-colored graph is a {\it total proper path} if $(i)$ any two adjacent edges on the path differ in color, $(ii)$ any two…
A tree $T$ in an edge-colored graph is a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be a fixed integer with $2\le k\le n$. For a vertex subset $S \subseteq…
In this paper, we investigate rainbow connection number $rc(G)$ of bridgeless outerplanar graphs $G$ with diameter 2 or 3. We proved the following results: If $G$ has diameter $2,$ then $rc(G)=3$ for fan graphs $F_{n}$ with $n\geq 7$ or…
A vertex-coloring of a connected graph $G$ is a strong conflict-free vertex-connection coloring if every two distinct vertices are joined by a shortest path on which some color appears exactly once. The minimum number of colors in such a…
Let $\mathcal G$ be a hypergraph whose edges are colored. An {\it $(\alpha,n)$-detachment} of $\mathcal G$ is a hypergraph obtained by splitting a vertex $\alpha$ into $n$ vertices, say $\alpha_1,\dots,\alpha_n$, and sharing the incident…