Dynamic Coloring of Graphs

A dynamic k-coloring of a graphG is a proper k-coloring of the vertices ofG such that every vertex of degree at least 2 in G will be adjacent to vertices with at least two different colors. The smallest number k for which a graph G has a dynamic k-coloring is the dynamic chromatic number χd(G). In this paper, we investigate the behavior of χd(G), the bounds for χd(G), the comparison between χd(G) and χ(G), the chromatic number of G, and other related problems and generalizations. This paper proposes the idea of a dynamic coloring of a graph, which we now motivate and define. A graph coloring assigns distinct colors to adjacent vertices. A coloring of the graph in which a typical vertex is adjacent to more than one color class represents a situation in which the typical individual has a greater variety in the types of relationships. Thus, the overall interactions would not be so limited but more dynamic. Hence, a dynamic coloring is defined as a proper coloring for which any vertex of degree at least two is adjacent to more than one color class. As in the case of graph colorings, a chromatic number for dynamic colorings may similarly be defined. Of course, any interesting problems of proper colorings may be reconsidered for dynamic colorings– the additional condition gives a fresh perspective. And the new condition creates significant problems that are unique for dynamic colorings. 1 Definitions, Terminology, and Notation In this section, the definitions, terminology, and notation used throughout are described. Terminology especially relevant to a particular section but not to previous sections will first be described in that section. With few exceptions, the definitions, terminology, and notation used here is consistent with that found in [2]. All graphs are finite and simple graphs. A graph with just one vertex is called trivial. Any graph is assumed to be connected, although the removal of edges or vertices may result in a graph that is not connected. A graph is typically denoted by G with vertex set V = V (G) and edge set E = E(G), with n denoting |V |. The neighbor set N(v) of a vertex v is the set of vertices adjacent to v. The degree of v is denoted by d(v). The minimum vertex degree is denoted by δ = δ(G), and the maximum vertex degree is denoted by