A coloring of the vertices of a graph G is said to be distinguishing provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, χD(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring....

We consider vertex coloring of an acyclic digraph ~ G in such a way that two vertices which have a common ancestor in ~ G receive distinct colors. Such colorings arise in a natural way when clustering, indexing and bounding space for various genetic data for efficient analysis. We discuss the corresponding chromatic number and derive an upper bound as a function of the maximum number of descend...

We consider vertex coloring of an acyclic digraph ~ G in such a way that two vertices which have a common ancestor in ~ G receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding chromatic number and derive an upper bound as a function of the maximum number of descendants of a given vertex an...

A coloring of a graph is an assignment of labels to certain elements of a graph. More commonly, elements are either vertices (vertex coloring), edges (edge coloring), or both edges and vertices (total colorings). The most common form asks to color the vertices of a graph such that no two adjacent vertices share the same “color” (label). This is called a proper vertex coloring. For a graph G, th...

This paper presents an analysis of solutions of the Graph Coloring Problem (GCP). Given an undirected graph G(V,E) with a vertex set V and an edge set E, the goal of GCP is to find a color assignment to every vertex in V such that any pair of adjacent (or connected) vertices receive different colors, and the total number of colors required for the feasible color assignment be minimized; the sma...

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. We obtain a new lower bound for the harmonious chromatic number of general graphs, in terms of the independence number of the graph, generalizing results of Moser [2].

The b-chromatic number of a graph G, denoted by b(G), is the largest positive integer k such that there exists a proper coloring for G with k colors in which every color class contains at least one vertex adjacent to some vertex in each of the other color classes, such a vertex is called a dominant vertex. The f -chromatic vertex number of a d-regular graph G, denoted by f(G), is the maximum nu...

An acyclic edge colouring of a graph is a proper edge colouring having no 2-coloured cycle, that is, a colouring in which the union of any two colour classes forms a linear forest. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge colouring using k colours and is usually denoted by a′(G). Determining a ′(G) exactly is a very hard problem (both the...

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 is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a(G) ≤ ∆ + 2, where ∆ = ∆(G) denotes the maximum degree of the gra...

let g be a simple graph and (g,) denotes the number of proper vertex colourings of gwith at most  colours, which is for a fixed graph g , a polynomial in  , which is called thechromatic polynomial of g . using the chromatic polynomial of some specific graphs, weobtain the chromatic polynomials of some nanostars.