Related papers: Distance domatic numbers for grid graphs

Given a set D of positive integers, the associated distance graph on the integers is the graph with the integers as vertices and an edge between distinct vertices if their difference lies in D. We investigate the chromatic numbers of…

A 2-distance k-coloring of a graph G is a mapping from V (G) to the set of colors {1,. .. , k} such that every two vertices at distance at most 2 receive distinct colors. The 2-distance chromatic number $\chi$ 2 (G) of G is then the mallest…

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 k-distance r-coloring of a graph is a coloring of the vertices of the graph such that if the distance between 2 vertices x and y is less or equal to k, then x and y must have distinct colors. A planar graph is a graph that can be drawn…

A vertex coloring of a graph $G$ is an assignment of colors to the vertices of $G$ such that every two adjacent vertices of $G$ have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the…

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…

We say a proper coloring of a graph is distance-$k$ fall if every vertex is within distance $k$ of at least one vertex of every color. We show that if $G$ is a connected graph of order at least $3$ that is $3$-colorable, thenit has a…

A 2-distance $k$-coloring of a graph $G$ is a proper $k$-coloring such that any two vertices at distance two or less get different colors. The 2-distance chromatic number of $G$ is the minimum $k$ such that $G$ has a 2-distance…

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 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 vertex coloring is called distinguishing if the identity is the only automorphism that can preserve it. The distinguishing threshold $\theta(G)$ of a graph $G$ is the minimum number of colors $k$ required that any arbitrary $k$-coloring…

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…

A 2-distance $k$-coloring of a graph $G$ is a proper $k$-coloring such that any two vertices at distance two or less get different colors. The 2-distance chromatic number of $G$ is the minimum $k$ such that $G$ has a 2-distance…

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 b-coloring of a graph is a proper coloring of its vertices such that each color class contains a vertex adjacent to at least one vertex of every other color class. The b-chromatic number of a graph is the largest integer k such that the…

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

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…

A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number…

A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic…

A graph is k-total colourable if there is an assignment of k different colours to the vertices and edges of the graph such that no two adjacent nor incident elements receive the same colour. The total chromatic number of some direct product…