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 belong 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 a new upper bound for the chromatic number of a graph in terms of its maximum average degree and show that this bound is an improvement on previous bounds. From this, we show a relationship between the average degree and the chromatic number of an average hereditary graph. This class of graphs is explored further by proving some interesting properties regarding the class of average hereditary graphs. An equivalent condition is provided for a graph to be average hereditary, through which we show that we can decide if a given graph is average hereditary in polynomial time. We then provide a construction 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.

翻译：在本文中，我们引入了一类称为平均遗传图的图。大多数在常规图论应用中出现的图均属于此类图。许多常见的图类型，如正则图、树及其他流行图类，均归属于此类。我们证明了图的最大平均度与其色数之间的一个新上界，并表明该界优于先前的结果。由此，我们揭示了平均遗传图的平均度与色数之间的关联。通过证明此类图的一些有趣性质，我们对平均遗传图类进行了进一步探索。我们给出了图成为平均遗传图的等价条件，据此可证明在多项式时间内判断给定图是否为平均遗传图。随后，我们提供了一种平均遗传图的构造方法，通过该方法可递归地构造平均遗传图。我们还证明了此类图在二元运算下封闭，由此获得了平均遗传图的另一种构造方法，并观察到此类图具有一些有趣的代数性质。最后，我们探讨了当输入限制为平均遗传图时，图3-着色问题的复杂度所受的影响。