Let G be a graph and χaa(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 χaa(G□H) for any two graphs G and H. We also determine exact value of this parameter for the Cartesian product of two paths, Cartesian product of a path and a cycle, C...

In this paper, we prove that a cubic line graph G on n vertices rather than the complete graph K4 has b3c vertex-disjoint triangles and the vertex independence number b3c. Moreover, the equitable chromatic number, acyclic chromatic number and bipartite density of G are 3, 3, 79 respectively.

Let G = (V,E) be any finite simple graph. A mapping C : E → [k] is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced by all the edges which have either colour i or j is acyclic. The smallest number k of colours, such that G has an acy...

Given a graph G, an automorphic edge(vertex)-coloring of G is a proper edge(vertex)-coloring such that each automorphism of the graph preserves the coloring. The automorphic chromatic index (number) is the least integer k for which G admits an automorphic edge(vertex)coloring with k colors. We show that it is NP-complete to determine the automorphic chromatic index and the automorphic chromatic...

An acyclic coloring of a graph G is a proper coloring of the vertex set of G such that G contains no bichromatic cycles. The acyclic chromatic number of a graph G is the minimum number k such that G has an acyclic coloring with k colors. In this paper, acyclic colorings of Hamming graphs, products of complete graphs, are considered.

The total chromatic number, Z"(G), of a graph G, is defined to be the minimum number ofcolours needed to colour the vertices and edges of a graph in such a way that no adjacent vertices, no adjacent edges and no incident vertex and edge are given the same colour. This paper shows that )('(G) _< z'(G) + 2x/~G), where z(G)is the vertex chromatic number and )((G)is the edge chromatic number of the...

An acyclic coloring of a graph G is a coloring of its vertices such that:(i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G. Recently it has been proved that any graph of maximum degree 5 has an acyclic chromatic number at most 8. In this paper we prese...

A labeling f : V (G) → {1, 2, . . . , d} of the vertex set of a graph G is said to be proper d-distinguishing if it is a proper coloring of G and any nontrivial automorphism of G maps at least one vertex to a vertex with a different label. The distinguishing chromatic number of G, denoted by χD(G), is the minimum d such that G has a proper d-distinguishing labeling. Let χ(G) be the chromatic nu...

For an arbitrary invariant ρ(G) of a graph G, the ρ−vertex stability number vsρ(G) is minimum vertices G whose removal results in H⊆G with ρ(H)≠ρ(G) or E(H)=∅. In this paper, first we give some general lower and upper bounds for ρ-vertex number, then study edge chromatic vertex graphs, vsχ′(G), where χ′=χ′(G) (chromatic index) G. We prove parameter determine vsχ′(G) specific classes graphs.

Let G be a simple graph. A total coloring f of G is called E-total-coloring if no two adjacent vertices of G receive the same color and no edge of G receives the same color as one of its endpoints. For E-total-coloring f of a graph G and any vertex u of G, let Cf (u) or C(u) denote the set of colors of vertex u and the edges incident to u. We call C(u) the color set of u. If C(u) 6= C(v) for an...