Related papers: Proper vertex connection and graph operations
A vertex-colored graph $G$ is {\it rainbow vertex-connected} if any pair of vertices in $G$ are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow…
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…
For a connected graph, we define the proper-walk connection number as the minimum number of colors needed to color the edges of a graph so that there is a walk between every pair of vertices without two consecutive edges having the same…
The proper connection number $pc(G)$ of a connected graph $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one path in $G$ such that no two…
This paper continues the study of a new variant of graph coloring with a connectivity constraint recently introduced by Hsieh et al. [COCOON 2024]. A path in a vertex-colored graph is called conflict-free if there is a color that appears…
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…
A vertex-colored graph is {\it rainbow vertex-connected} if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection} of a…
A path in an(a) edge(vertex)-colored graph is called a conflict-free path if there exists a color used on only one of its edges(vertices). An(A) edge(vertex)-colored graph is called conflict-free (vertex-)connected if for each pair of…
A proper vertex coloring of a graph is a mapping of its vertices on a set of colors, such that two adjacent vertices are not mapped to the same color. This constraint may be interpreted in terms of the distance between to vertices and so a…
Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color can appear with less than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper…
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 by $rc(G)$, is the smallest number of colors that…
A path in an edge-coloured graph is called \emph{rainbow path} if its edges receive pairwise distinct colours. An edge-coloured graph is said to be \emph{rainbow connected} if any two distinct vertices of the graph are connected by a…
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…
An edge-coloured path is monochromatic if all of its edges have the same colour. For a $k$-connected graph $G$, the monochromatic $k$-connection number of $G$, denoted by $mc_k(G)$, is the maximum number of colours in an edge-colouring of…
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…
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…
A path $P$ in an edge-colored graph $G$ is a \emph{proper path} if no two adjacent edges of $P$ are colored with the same color. The graph $G$ is \emph{proper connected} if, between every pair of vertices, there exists a proper path in $G$.…
A tree $T$ in an edge-colored graph is called 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 an integer with $2\leq k \leq n$. For $S\subseteq V(G)$ and $|S|…
A path in an edge-colored graph $G$ is called monochromatic if any two edges on the path have the same color. For $k\geq 2$, an edge-colored graph $G$ is said to be monochromatic $k$-edge-connected if every two distinct vertices of $G$ are…
Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the…