Related papers: Distance-two coloring of sparse graphs

In this paper we introduce the notion of $\Sigma$-colouring of a graph $G$: For given subsets $\Sigma(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that…

Given a graph $G$ and a list assignment $L(v)$ for each vertex of $v$ of $G$. A proper $L$-list-coloring of $G$ is a function that maps every vertex to a color in $L(v)$ such that no pair of adjacent vertices have the same color. We say…

Given a graph $G$ and an integer $p$, a coloring $f : V(G) \to \mathbb{N}$ is \emph{$p$-centered} if for every connected subgraph $H$ of $G$, either $f$ uses more than $p$ colors on $H$ or there is a color that appears exactly once in $H$.…

Let $G$ be a graph and $f:V(G)\rightarrow \mathbb{N}$ be a function. An $f$-coloring of a graph $G$ is an edge coloring such that each color appears at each vertex $v\in V(G)$ at most $f (v)$ times. The minimum number of colors needed to…

A 2-distance list k-coloring of a graph is a proper coloring of the vertices where each vertex has a list of at least k available colors and vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance…

A vertex coloring of a graph G is called a 2-distance coloring if any two vertices at a distance at most 2 from each other receive different colors. Suppose that G is a planar graph with a maximum degree at most 5. We prove that G admits a…

A $2$-distance $k$-coloring of a graph is a proper $k$-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance ($\Delta+2$)-coloring for graphs with maximum average…

A vertex coloring of a graph $G$ is called a 2-distance coloring if any two vertices at a distance at most $2$ from each other receive different colors. Suppose that $G$ is a planar graph with girth $5$ and maximum degree $\Delta$. We prove…

In this paper, we consider coloring of graphs under the assumption that some vertices are already colored. Let $G$ be an $r$-colorable graph and let $P\subset V(G)$. Albertson [J.\ Combin.\ Theory Ser. B \textbf{73} (1998), 189--194] has…

Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of non-empty sets. By an $L$-coloring of $G$ we mean a (proper) coloring $\phi$ of $G$ such that $\phi(v)\in L(v)$ for every vertex $v$ of $G$. Thomassen…

For a graph $G$ and $t,k\in\mathbb{Z}^+$ a \emph{$t$-tone $k$-coloring} of $G$ is a function $f:V(G)\rightarrow \binom{[k]}{t}$ such that $|f(v)\cap f(w)| < d(v,w)$ for all distinct $v,w \in V(G)$. The \emph{$t$-tone chromatic number} of…

A vertex coloring of a graph $G$ is said to be a 2-distance coloring if any two vertices at distance at most $2$ from each other receive different colors, and the least number of colors for which $G$ admits a $2$-distance coloring is known…

A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors…

Conflict-free coloring (in short, CF-coloring) of a graph $G = (V,E)$ is a coloring of $V$ such that the neighborhood of each vertex contains a vertex whose color differs from the color of any other vertex in that neighborhood. Bounds on…

Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an…

We consider two graph colouring problems in which edges at distance at most $t$ are given distinct colours, for some fixed positive integer $t$. We obtain two upper bounds for the distance-$t$ chromatic index, the least number of colours…

A proper $k$-coloring of a graph $G$ is a \emph{neighbor-locating $k$-coloring} if for each pair of vertices in the same color class, the two sets of colors found in their respective neighborhoods are different. The…

Given a graph $G$ and a non-decreasing sequence $S=(a_1,a_2,\ldots)$ of positive integers, the mapping $f:V(G) \rightarrow \{1,\ldots,k\}$ is an $S$-packing $k$-coloring of $G$ if for any distinct vertices $u,v\in V(G)$ with $f(u)=f(v)=i$…

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$-admissible if admits a spanning tree in which the distance between any two adjacent vertices of $G$…

List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a…