Related papers: Colouring graphs with constraints on connectivity

Let $\mathcal{C}_k(n)$ be the family of all connected $k$-chromatic graphs of order $n$. Given a natural number $x\geq k$, we consider the problem of finding the maximum number of $x$-colorings among graphs in $\mathcal{C}_k(n)$. When…

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…

A complete $k$-coloring of a graph $G=(V,E)$ is an assignment $\varphi:V\to\{1,\ldots,k\}$ of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one…

An edge-colored graph is said to be balanced if it has an equal number of edges of each color. Given a graph $G$ whose edges are colored using two colors and a positive integer $k$, the objective in the Edge Balanced Connected Subgraph…

Let $P_t$ and $C_\ell$ denote a path on $t$ vertices and a cycle on $\ell$ vertices, respectively. In this paper we study the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved…

A graph $H$ is {\em $p$-edge colorable} if there is a coloring $\psi: E(H) \rightarrow \{1,2,\dots,p\}$, such that for distinct $uv, vw \in E(H)$, we have $\psi(uv) \neq \psi(vw)$. The {\sc Maximum Edge-Colorable Subgraph} problem takes as…

A conflict-free $k$-coloring of a graph $G=(V,E)$ assigns one of $k$ different colors to some of the vertices such that, for every vertex $v$, there is a color that is assigned to exactly one vertex among $v$ and $v$'s neighbors. Such…

For $k\in \mathbb{N}$, a $k$-acyclic colouring of a graph $G$ is a function $f\colon V(G)\to \{0,1,\dots,k-1\}$ such that (i)~$f(u)\neq f(v)$ for every edge $uv$ of $G$, and (ii)~there is no cycle in $G$ bicoloured by $f$. For $k\in…

Given a set of obstacles and two points, is there a path between the two points that does not cross more than $k$ different obstacles? This is a fundamental problem that has undergone a tremendous amount of work. It is known to be NP-hard,…

We study how the complexity of the graph colouring problems star colouring and restricted star colouring vary with the maximum degree of the graph. Restricted star colouring (in short, rs colouring) is a variant of star colouring. For $k\in…

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…

We continue research into a well-studied family of problems that ask whether the vertices of a graph can be partitioned into sets $A$ and~$B$, where $A$ is an independent set and $B$ induces a graph from some specified graph class ${\cal…

Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a…

Let $G$ be a connected graph with minimum degree $\delta(G)$ and vertex-connectivity $\kappa(G)$. The graph $G$ is $k$-connected if $\kappa(G)\geq k$, maximally connected if $\kappa(G) = \delta(G)$, and super-connected (or super-$\kappa$)…

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 each vertex has an equal number of neighbors of each color is called neighborhood-balanced…

The semistrong edge coloring, as a relaxation of the well-known strong edge coloring, can be used to model efficient communication scheduling in wireless networks. An edge coloring of a graph $G$ is called \emph{semistrong} if every color…

Let $\mathbf{k} := (k_1,\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges…

For graphs $G$ and $H$, an $H$-colouring of $G$ is a map $\psi:V(G)\rightarrow V(H)$ such that $ij\in E(G)\Rightarrow\psi(i)\psi(j)\in E(H)$. The number of $H$-colourings of $G$ is denoted by $\hom(G,H)$. We prove the following: for all…

We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. This is one of the most challenging problems in graph algorithms. In this paper using Blum's notion of ``progress'', we develop 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…