A $k$-proper edge-coloring of a graph G is called adjacent vertex-distinguishing if any two vertices are distinguished by the set colors appearing in edges incident to each vertex. The smallest value $k$ for which $G$ admits such coloring denoted $\chi'_a(G)$. We prove that $\chi'_a(G) = 2R + 1$ most circulant graphs $C_n([1, R])$.

To attack the Four Color Problem, in 1880, Tait gave a necessary and sufficient condition for plane triangulations to have a proper 4-vertex-coloring: a plane triangulation G has a proper 4-vertex-coloring if and only if the dual of G has a proper 3-edge-coloring. A cyclic coloring of a map G on a surface F 2 is a vertex-coloring of G such that any two vertices x and y receive different colors ...

A twin edge k-coloring of a graph G is a proper edge k-coloring of G with the elements of Zk so that the induced vertex k-coloring, in which the color of a vertex v in G is the sum in Zk of the colors of the edges incident with v, is a proper vertex k-coloring. The minimum k for which G has a twin edge k-coloring is called the twin chromatic index of G. Twin chromatic index of the square P 2 n ...

The motivation of this paper comes from statistical physics as well as from combinatorics and topology. The general setup in statistical mechanics can be outlined as follows. Let G be a graph and let C be a finite set of “states” or “colors”. We think of G as a crystal in which either the edges or the vertices are regarded as “sites” which can have states from C. In the first case we speak abou...

An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a(G), is the minimum number of colors required for acyclic vertex coloring of graph G = (V,E). For a family F of graphs, the acyclic chromatic number of F , denoted by a(F), is defined as the maximum a(G) over all the graphs G ∈ F . In this pape...

An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a(G), is the minimum number of colors required for acyclic vertex coloring of a graph G = (V,E). For a family F of graphs, the acyclic chromatic number of F , denoted by a(F ), is defined as the maximum a(G) over all the graphs G ∈ F . In this p...

Defective coloring is a variant of the traditional vertex-coloring in which adjacent vertices are allowed to have the same color, as long as the induced monochromatic components have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, diameter, and acyc...

In this paper we present a polynomial time algorithm that determines if an input graph containing no induced seven-vertex path is 3-colorable. This affirmatively answers a question posed by Randerath, Schiermeyer and Tewes in 2002. Our algorithm also solves the list-coloring version of the 3-coloring problem, where every vertex is assigned a list of colors that is a subset of {1, 2, 3}, and giv...

A defining set (of vertex coloring) of a graph G is a set of vertices S with an assignment of colors to its elements which has a unique completion to a proper coloring of G. We define a minimal defining set to be a defining set which does not properly contain another defining set. If G is a uniquely vertex colorable graph, clearly its minimum defining sets are of size χ(G) − 1. It is shown that...

Given a graph G, its k-coloring graph is the graph whose vertex set is the proper k-colorings of the vertices of G with two k−colorings adjacent if they differ at exactly one vertex. In this paper, we consider the question: Which graphs can be coloring graphs? In other words, given a graph H, do there exist G and k such that H is the k-coloring graph of G? We will answer this question for sever...