A study on dominator coloring of Various families of graphs

Abstract

Graph theory is becoming increasingly significant as it is used newlineto study and prepare structural models of objects or technologies that newlinelead to new inventions or modifications in the existing environment, for newlinedevelopment in various fields. newlineGraph coloring and domination are the two popular and newlineextensively researched areas in the field of graph theory. Coloring newlinerepresented by and#119891;: and#119881; and#8594; and#119862; is a mapping from the set of nodes and#119881; of a graph newlineand#119866; to a set of colors and#119862;. A proper coloring is one in which no two adjacent newlinenodes are assigned the same color. The smallest number of colors newlinerequired in any proper coloring of a graph and#119866; is defined as the chromatic newlinenumber of and#119866; and is denoted by and#120594;(and#119866;). newlineA subset and#119863;and#119878; of the node set and#119881; of a graph and#119866; is called the newlinedominating set of the graph if every node in and#119881; and#8726; and#119863;and#119878; is adjacent to a newlinenode in and#119863;and#119878;. The domination number denoted by and#120574;(and#119866;), is the minimum newlinecardinality of a dominating set in and#119866;. newlineThe dominating set and coloring of graphs have numerous newlineapplications, and this has led to the study of several variants of these newlineproblems. One such variant which combines the methodology of newlinedomination and node coloring of graphs is known as dominator coloring newlineof graphs. This notion was introduced by Hedetniemi and further studied newlineby Ralucca Michelle Gera in 2006. The dominator coloring of a graph is newlinedefined as a proper coloring of nodes in which each node of the graph newlinedominates all nodes of at least a color class. Thus, the methodology used newlinevii newlineto determine the dominator coloring number of a graph and#119866; denoted by newlineand#120594;and#119889; (and#119866;), is to find the minimum number of colors assigned to the nodes newlineof G in such a way that each node of the graph is either a neighbor to newlineevery node of some (other) color class or is a member of a singleton newlinecolor class. newlineThere has been a lot of interest in the research of dominator newlinecoloring of graphs and good research publications have been done on it. newlineAlthough many results on dominator coloring have been published, newlinedominator coloring has to be explored for many classes of graphs