Related papers: Injective coloring of graphs revisited

An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor have distinct colors. A graph $G$ is injectively $k$-choosable if for any list assignment $L$, where…

A proper vertex coloring of a connected graph $G$ is called an odd coloring if, for every vertex $v$ in $G$, there exists a color that appears odd number of times in the open neighborhood of $v$. The minimum number of colors required to…

The chromatic index $\chi'(G)$ of a graph $G$ is the smallest $k$ for which $G$ admits an edge $k$-coloring such that any two adjacent edges have distinct colors. The strong chromatic index $\chi'_s(G)$ of $G$ is the smallest $k$ such that…

A $k$-subcolouring of a graph $G$ is a function $f:V(G) \to \{0,\ldots,k-1\}$ such that the set of vertices coloured $i$ induce a disjoint union of cliques. The subchromatic number, $\chi_{\textrm{sub}}(G)$, is the minimum $k$ such that $G$…

For a bipartite graph $G$ with parts $X$ and $Y$, an $X$-interval coloring is a proper edge coloring of $G$ by integers such that the colors on the edges incident to any vertex in $X$ form an interval. Denote by $\chi'_{int}(G,X)$ the…

A set $P$ of vertices in a graph $G$ is an open packing if no two distinct vertices in $P$ have a common neighbor. Among all maximal open packings in $G$, the smallest cardinality is denoted $\rho^{\rm o}_L(G)$ and the largest cardinality…

A graph is called uniquely distinguishing colorable if there is only one partition of vertices of the graph that forms distinguishing coloring with the smallest possible colors. In this paper, we study the unique colorability of the…

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…

Two relationships between the injective chromatic number and, respectively, chromatic number and chromatic index, are proved. They are applied to determine the injective chromatic number of Sierpi\'nski graphs and to give a short proof that…

Indicated coloring is a type of game coloring in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly,…

For a graph $G$, let $\nu_s(G)$ be the induced matching number of $G$. We prove the sharp bound $\nu_s(G)\geq \frac{n(G)}{9}$ for every graph $G$ of maximum degree at most $4$ and without isolated vertices that does not contain a certain…

The strong chromatic number $\chi_{\text{s}}(G)$ of a graph $G$ on $n$ vertices is the least number $r$ with the following property: after adding $r \lceil n/r \rceil - n$ isolated vertices to $G$ and taking the union with any collection of…

A $k$-coloring of a graph $G$ is a $k$-partition $\Pi=\{S_1,\ldots,S_k\}$ of $V(G)$ into independent sets, called \emph{colors}. A $k$-coloring is called \emph{neighbor-locating} if for every pair of vertices $u,v$ belonging to the same…

A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges of distance at most two receive distinct colors. The minimum number of colors we need in order to give $G$ a strong edge-coloring is called the strong…

An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval…

Let $G$ be a simple graph. The dominated coloring of a graph $G$ is a proper coloring of $G$ such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of $G$ is called the…

A strong edge-coloring of a graph $G$ is an edge-coloring such that no two edges of distance at most two receive the same color. The strong chromatic index $\chi'_s(G)$ is the minimum number of colors in a strong edge-coloring of $G$. P.…

For $k\ge 1$, we consider interleaved $k$-tuple colorings of the nodes of a graph, that is, assignments of $k$ distinct natural numbers to each node in such a way that nodes that are connected by an edge receive numbers that are strictly…

A conflict-free k-coloring of a graph 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 colorings have…

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…