Related papers: New Bounds for the Acyclic Chromatic Index

The smallest integer $k$ needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph. Vizing and Behzed conjectured that the total coloring can be…

Let $k \ge 1$ be an integer and let $G$ be a nonempty simple graph. An \emph{edge-$k$-coloring} $\varphi$ of $G$ is an assignment of colors from $\{1,\ldots,k\}$ to the edges of $G$ such that no two adjacent edges receive the same color.…

An adjacent vertex distinguishing edge-coloring or an \avd-coloring of a simple graph $G$ is a proper edge-coloring of $G$ such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree…

The distinguishing index $D'(G)$ of a graph $G$ is the least number of colors necessary to obtain an edge coloring of $G$ that is preserved only by the trivial automorphism. We show that if $G$ is a connected $\alpha$-regular graph for some…

Following a given ordering of the edges of a graph $G$, the greedy edge colouring procedure assigns to each edge the smallest available colour. The minimum number of colours thus involved is the chromatic index $\chi'(G)$, and the maximum…

In this paper, we introduce the notion of $t$-tone edge coloring. A $t$-tone edge $k$-coloring of a graph $G$ assigns to each edge of $G$ a set of $t$ distinct colors from $\{1,\dots,k\}$ such that any two edges at distance $d$ share fewer…

This paper studies two variants of defective acyclic coloring of planar graphs. For a graph $G$ and a coloring $\varphi$ of $G$, a 2CC transversal is a subset $E'$ of $E(G)$ that intersects every 2-colored cycle. Let $k$ be a positive…

An \emph{interval $t$-coloring} of a graph $G$ is a proper edge-coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A graph $G$ is called \emph{interval…

A strong edge-coloring of a graph $G$ is an edge-coloring in which every color class is an induced matching, and the strong chromatic index $\chi_s'(G)$ is the minimum number of colors needed in strong edge-colorings of $G$. A graph is…

We say that an edge colouring breaks an automorphism if some edge is mapped to an edge of a different colour. We say that the colouring is distinguishing if it breaks every non-identity automorphism. We show that such colouring can be…

For an oriented graph $G$, let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$. Let $f(n)$ be the smallest integer such that every oriented graph $G$ with chromatic number larger than $f(n)$ has $f(G) > n$. Let…

A tree $T$, in an edge-colored graph $G$, is called {\em a rainbow tree} if no two edges of $T$ are assigned the same color. A {\em $k$-rainbow coloring}of $G$ is an edge coloring of $G$ having the property that for every set $S$ of $k$…

The dichromatic and diachromatic numbers of a digraph are the minimum and maximum numbers of colors, respectively, in acyclic and complete colorings of the digraph. In this paper, we construct, for all $r \leq t$, non-symmetric digraphs…

Let $G$ be a graph and $R\subseteq V(G)$. A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an $R$-sequential $t$-coloring if the edges incident to each vertex $v\in R$ are colored by the colors $1,\ldots,d_{G}(v)$,…

A well-known theorem of Vizing states that if $G$ is a simple graph with maximum degree $\Delta$, then the chromatic index $\chi'(G)$ of $G$ is $\Delta$ or $\Delta+1$. A graph $G$ is class 1 if $\chi'(G)=\Delta$, and class 2 if…

A signed graph is a pair $(G, \sigma)$, where $G$ is a graph and $\sigma: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend…

A strong edge coloring of a graph $G$ is an assignment of colors to the edges of $G$ such that two distinct edges are colored differently if they are incident to a common edge or share an endpoint. The strong chromatic index of a graph $G$,…

A vertex coloring of a graph is said to be \textit{conflict-free} with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al.,…

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…

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $G$ is a proper edge coloring of $G$ such that each color class is dominated by at least one edge of $G$. The minimum number of colors among…