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We consider edge colorings of graphs. An edge coloring is a majority coloring if for every vertex at most half of the edges incident with it are in one color. And edge coloring is a distinguishing coloring if for every non-trivial…

A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,...,t$ such that each of $t$ colors is used, and adjacent edges are colored differently. The set of colors of edges incident with a vertex $x$ of $G$…

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which…

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

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph $G$ is decomposable into $k$ odd subgraphs if its edge set can be partitioned into $k$ subsets each of which induces an odd subgraph of $G$. The minimum…

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that no path or cycle of length four is bi-colored. The star chromatic index of $G$, denoted by $\chi^{\prime}_{s}(G)$, is the minimum $k$ such that $G$ admits a star…

We call a proper edge coloring of a graph $G$ a B-coloring if every 4-cycle of $G$ is colored with four different colors. Let $q_B(G)$ denote the smallest number of colors needed for a B-coloring of $G$. Motivated by earlier papers on…

Let $G$ be an edge-coloured graph. The minimum colour degree $\delta^c(G)$ of $G$ is the largest integer $k$ such that, for every vertex $v$, there are at least $k$ distinct colours on edges incident to $v$. We say that $G$ is properly…

A $(1,1,2,2)$-coloring of a graph is a partition of its vertex set into four sets two of which are independent and the other two are $2$-packings. In this paper, we prove that every claw-free cubic graph admits a $(1,1,2,2)$-coloring. This…

A strong edge-coloring of a graph $G=(V,E)$ is a partition of its edge set $E$ into induced matchings. We study bipartite graphs with one part having maximum degree at most $3$ and the other part having maximum degree $\Delta$. We show that…

An edge-coloring of a multigraph G with colors 1,2,...,t is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In this paper we prove that…

A \emph{star coloring} of a graph $G$ is a proper vertex-coloring such that no path on four vertices is $2$-colored. The minimum number of colors required to obtain a star coloring of a graph $G$ is called star chromatic number and it is…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and…

Proper conflict-free coloring is an intermediate notion between proper coloring of a graph and proper coloring of its square. It is a proper coloring such that for every non-isolated vertex, there exists a color appearing exactly once in…

The acyclic chromatic index (or acyclic edge-chromatic number) of a graph is the least number of colors needed to properly color its edges so that none of its cycles has only two colors. We show that for a graph of max degree $\Delta$, the…

In the past various distance based colorings on planar graphs were introduced. We turn our focus to three of them, namely $2$-distance coloring, injective coloring, and exact square coloring. A $2$-distance coloring is a proper coloring of…

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…

The star chromatic index of a multigraph $G$, denoted by $\chi_{\mathrm{star}}'(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length $4$ is bicolored. In this paper, we study…