On graphs with well distributed edge density

In this paper we introduce a class of graphs which we call average hereditary graphs. Most graphs that occur in the usual graph theory applications belongs to this class of graphs. Many popular types of graphs fall under this class, such as regular graphs, trees and other popular classes of graphs. We prove an upper bound for the chromatic number of average hereditary graphs, and show that this bound is an improvement on previous bounds. This class of graphs is explored further by proving some interesting properties regarding the class of average hereditary graphs. We analyze the computational complexity of deciding if an arbitrary graph is average hereditary. Then an equivalent condition and a polynomial time sufficient condition is provided for a graph to be average hereditary. We then provide a constructions for average hereditary graphs, using which an average hereditary graph can be recursively constructed. We also show that this class of graphs is closed under a binary operation, from this another construction is obtained for average hereditary graphs, and we see some interesting algebraic properties this class of graphs has. We then explore the effect on the complexity of graph 3-coloring problem when the input is restricted to average hereditary graphs.