Related papers: Connectedness of Strong k-Colour Graphs
A path in a vertex-colored graph is a {\it vertex-proper path} if any two internal adjacent vertices differ in color. A vertex-colored graph is {\it proper vertex $k$-connected} if any two vertices of the graph are connected by $k$ disjoint…
A path in a vertex-colored graph is a {\it vertex-proper path} if any two internal adjacent vertices differ in color. A vertex-colored graph is {\it proper vertex $k$-connected} if any two vertices of the graph are connected by $k$ disjoint…
Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V(G) into disjoint sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex…
For a simple graph G = (V, E) and a positive integer k greater than or equal to 2, a coloring of vertices of G using exactly k colors such that every vertex has an equal number of vertices of each color in its closed neighborhood is called…
An edge-colored graph $G$ is $k$-color connected if, between each pair of vertices, there exists a path using at least $k$ different colors. The $k$-color connection number of $G$, denoted by $cc_{k}(G)$, is the minimum number of colors…
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…
The reconfiguration graph for the $k$-colourings of a graph $G$, denoted $R_{k}(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on exactly one vertex. For any…
A tree $T$ in an edge-colored graph is a \emph{proper tree} if any two adjacent edges of $T$ are colored with different colors. Let $G$ be a graph of order $n$ and $k$ be a fixed integer with $2\leq k\leq n$. For a vertex set $S\subseteq…
An edge coloring $c$ of a graph $G$ is a royal $k$-edge coloring of $G$ if the edges of $G$ are assigned nonempty subsets of the set $\{1, 2, \ldots, k\}$ in such a way that the vertex coloring obtained by assigning the union of the colors…
A subgraph $H$ of a multigraph $G$ is called strongly spanning, if any vertex of $G$ is not isolated in $H$, while it is called maximum $k$-edge-colorable, if $H$ is proper $k$-edge-colorable and has the largest size. We introduce a…
The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph $G$ contains as its vertex set the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on just one vertex of $G$. We show that for each…
A strong odd coloring of a simple graph $G$ is a proper coloring of the vertices of $G$ such that for every vertex $v$ and every color $c$, either $c$ is used an odd number of times in the open neighborhood $N_G(v)$ or no neighbor of $v$ is…
A uniquely $k$-colourable graph is a graph with exactly one partition of the vertex set into at most $k$ colour classes. Here, we investigate some constructions of uniquely $k$-colourable graphs and give a construction of $K_k$-free…
A vertex-colored graph $G$ is said to be rainbow vertex-connected if every two vertices of $G$ are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number…
For a graph $G$, the $k$-recolouring graph $\mathcal{R}_k(G)$ is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on exactly one vertex. We prove that for all $n \ge…
A proper coloring of a graph $G$ is said to be a strong odd coloring of $G$, if for every vertex $v$ and every color $c$, either $c$ appears on an odd number of vertices in the neighborhood of $v$ or $c$ is absent in the neighborhood of…
Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We…
Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$…
A fall $k$-coloring of a graph $G$ is a proper $k$-coloring of $G$ such that each vertex of $G$ sees all $k$ colors on its closed neighborhood. In this paper, we answer some questions of \cite{dun} about some relations between fall…
For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring…