‎let $g$ be a graph and $chi^{prime}_{aa}(g)$ denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of $g$ in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎we prove a general bound for $chi^{prime}_{aa}(gsquare h)$ for any two graphs $g$ and $h$‎. ‎we also determine‎ ‎exact value of this parameter for the cartesian product of ...

Let G = (V (G), E(G)) be a simple graph and T (G) be the set of vertices and edges of G. Let C be a k−color set. A (proper) total k−coloring f of G is a function f : T (G) −→ C such that no adjacent or incident elements of T (G) receive the same color. For any u ∈ V (G), denote C(u) = {f(u)} ∪ {f(uv)|uv ∈ E(G)}. The total k−coloring f of G is called the adjacent vertex-distinguishing if C(u) 6=...

An adjacent vertex distinguishing edge-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. The minimum number of colors χa(G) required to give G an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove χa(G) ≤ 5 for such graphs with maximum degree Δ(G) = 3 and prove χa(G) ≤ Δ(G) ...

In this paper, we study a new coloring parameter of graphs called the gap vertexdistinguishing edge coloring. It consists in an edge-coloring of a graph G which induces a vertex distinguishing labeling of G such that the label of each vertex is given by the difference between the highest and the lowest colors of its adjacent edges. The minimum number of colors required for a gap vertex-distingu...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $chi_a '(G)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. The maximum degree in $G$ denoted by $Delta(G)$, is the lower bound for $chi_a '(G)$. $P$-cuts introduced in this paper acts as a powerfu...

an acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. the acyclic chromatic index of a graph $g$ denoted by $chi_a '(g)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. the maximum degree in $g$ denoted by $delta(g)$, is the lower bound for $chi_a '(g)$. $p$-cuts introduced in this paper acts as a powerfu...

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G is the least number of colors in an acyclic edge coloring of G. In this paper, it is proved that the acyclic edge chromatic number of a planar graph G is at most ∆(G)+2 if G contains no i-cycles, 4≤ i≤ 8, or any two 3-cycles are not incident with a common vertex and ...

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G is the least number of colors in an acyclic edge coloring of G. In this paper, it is proved that the acyclic edge chromatic number of a planar graph G is at most ∆(G)+2 if G contains no i-cycles, 4≤ i≤ 8, or any two 3-cycles are not incident with a common vertex and ...

The adjacent vertex-distinguishing proper edge-coloring is the minimum number of colors required for a G, in which no two vertices are incident to edges colored with same set colors. an G called edge-chromatic index. In this paper, I compute index Anti-prism, sunflower graph, double triangular winged prism, rectangular prism and Polygonal snake graph.