Related papers: On Edge Coloring of Multigraphs

Let $mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $\chi_i(G)$ denote the injective chromatic number of $G$. We prove that if $mad(G) \leq 5/2$, then $\chi_i(G)\leq\Delta(G) + 1$; and if $mad(G) < 42/19$,…

A strong edge coloring of a graph is a proper edge coloring in which every color class is an induced matching. The strong chromatic index $\chi_s'(G)$ of a graph $G$ is the minimum number of colors in a strong edge coloring of $G$. Let…

A star edge coloring of a graph $G$ is a proper edge coloring with no 2-colored path or cycle of length four. The star edge coloring problem is to find an edge coloring of a given graph $G$ with minimum number $k$ of colors such that $G$…

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…

Given an arbitrary graph $G$ we study the chromatic number of a random subgraph $G_{1/2}$ obtained from $G$ by removing each edge independently with probability $1/2$. Studying $\chi(G_{1/2})$ has been suggested by Bukh~\cite{Bukh}, who…

A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a \emph{cyclic interval $t$-coloring} if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is…

The Grundy (or First-Fit) chromatic number of a graph $G=(V,E)$, denoted by $\Gamma(G)$ (or $\chi_{_{\sf FF}}(G)$), is the maximum number of colors used by a First-Fit (greedy) coloring of $G$. To determine $\Gamma(G)$ is NP-complete for…

A graph is perfect if the chromatic number of every induced subgraph equals the size of its largest clique, and an algorithm of Gr\"otschel, Lov\'asz, and Schrijver from 1988 finds an optimal colouring of a perfect graph in polynomial time.…

Let $G$ be a graph with chromatic number $\chi$, maximum degree $\Delta$ and clique number $\omega$. Reed's conjecture states that $\chi \leq \lceil (1-\varepsilon)(\Delta + 1) + \varepsilon\omega \rceil$ for all $\varepsilon \leq 1/2$. It…

The star chromatic index $\chi_s'(G)$ of a graph $G$ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored. We obtain a near-linear upper bound in terms of the…

Let $G$ be a simple graph with maximum degree $\Delta$. We call $G$ \emph{overfull} if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. The \emph{core} of $G$, denoted $G_{\Delta}$, is the subgraph of $G$ induced by its vertices of degree $\Delta$.…

If G is a graph and H is a set of subgraphs of G, then an edge-coloring of G is called H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, denoted poly_H(G), is the largest…

An acyclic edge coloring of a graph is a proper edge coloring without any bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$ colors.…

\textit{Total Coloring} of a graph is a major coloring problem in combinatorial mathematics, introduced in the early $1960$s. A \textit{total coloring} of a graph $G$ is a map $f:V(G) \cup E(G) \rightarrow \mathcal{K}$, where $\mathcal{K}$…

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…

Given a graph $G$, an injective edge-coloring of $G$ is a function $\psi:E(G) \rightarrow \mathbb N$ such that if $\psi(e) = \psi(e')$, then no third edge joins an endpoint of $e$ and an endpoint of $e'$. The injective chromatic index of a…

For a given graph $G$, the least integer $k\geq 2$ such that for every Abelian group $\mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)\rightarrow \mathcal{G}$ so that $\sum_{x\in N(u)}f(xu)\neq \sum_{x\in N(v)}f(xv)$…

The acyclic chromatic index of a graph $G$ is the least number of colors needed to properly color its edges so that none of its cycles is bichromatic. In this work, we show that $2\Delta-1$ colors are sufficient to produce such a coloring,…

An $(m,n)$-colored mixed graph $G$ is a graph with its arcs having one of the $m$ different colors and edges having one of the $n$ different colors. A homomorphism $f$ of an $(m,n)$-colored mixed graph $G$ to an $(m,n)$-colored mixed graph…

An edge coloring of a graph $G$ is called an acyclic edge coloring if it is proper and every cycle in $G$ contains edges of at least three different colors. The least number of colors needed for an acyclic edge coloring of $G$ is called the…