Related papers: Geometric achromatic and pseudoachromatic indices

A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance…

We show that the circular chromatic index of a (sub)cubic graph with girth at least six is at most 7/2.

Frank Harary introduced the concept of integral sum graph. A graph $G$ is an \emph{ integral sum graph} if its vertices can be labeled with distinct integers so that $e = uv$ is an edge of $G$ if and only if the sum of the labels on…

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number of a graph $G$ is the largest integer $k$ such that $G$ admits a…

A path in an edge-colored graph $G$ is called monochromatic if any two edges on the path have the same color. For $k\geq 2$, an edge-colored graph $G$ is said to be monochromatic $k$-edge-connected if every two distinct vertices of $G$ are…

A $k$-edge-colored graph is a finite, simple graph with edges labeled by numbers $1,\ldots,k$. A function from the vertex set of one $k$-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two…

The Additive Coloring Problem is a variation of the Coloring Problem where labels of $\{1,\ldots,k\}$ are assigned to the vertices of a graph $G$ so that the sum of labels over the neighborhood of each vertex is a proper coloring of $G$.…

The cochromatic number $\zeta(G)$ of a graph $G$ is the minimum number of colours needed for a vertex colouring where every colour class is either an independent set or a clique. Let $\chi(G)$ denote the usual chromatic number. Around 1991…

A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by $\chi_{s}'(G)$ which is the minimum number of colors that allow a…

A strong $k$-edge-coloring of a graph $G$ is a mapping from $E(G)$ to $\{1,2,\ldots,k\}$ such that every pair of distinct edges at distance at most two receive different colors. The strong chromatic index $\chi'_s(G)$ of a graph $G$ is the…

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 special spanning tree in which the distance between any two adjacent vertices…

Let $G$ be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let the color degree of a vertex $v$ be the number of different colors that are used on the…

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…

We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The…

In the paper we give a lower bound for the number of vertices of a given graph using its chromatic number. We find the graphs for which this bound is exact. The results are applied in the theory of Foklman numbers.

We determine (partly by computer search) the chromatic index (edge-chromatic number) of many strongly regular graphs (SRGs), including the SRGs of degree $k \leq 18$ and their complements, the Latin square graphs and their complements, and…

A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a…

Let $c:E(G)\to [k]$ be an edge-coloring of a graph $G$, not necessarily proper. For each vertex $v$, let $\bar{c}(v)=(a_1,\ldots,a_k)$, where $a_i$ is the number of edges incident to $v$ with color $i$. Reorder $\bar{c}(v)$ for every $v$ in…

An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$…

Recently, Behr introduced a notion of the chromatic index of signed graphs and proved that for every signed graph $(G$, $\sigma)$ it holds that \[ \Delta(G)\leq\chi'(G\text{, }\sigma)\leq\Delta(G)+1\text{,} \] where $\Delta(G)$ is the…