How to efficiently represent a graph in computer memory is a fundamental data structuring question. In the present paper, we address this question from a combinatorial point of view. A representation of an $n$-vertex graph $G$ is called implicit if it assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency of two vertices is a function of their codes. A necessary condition for a hereditary class $X$ of graphs to admit an implicit representation is that $X$ has at most factorial speed of growth. This condition, however, is not sufficient, as was recently shown in [Hatami & Hatami, FOCS 2022]. Several sufficient conditions for the existence of implicit representations deal with boundedness of some parameters, such as degeneracy or clique-width. In the present paper, we analyze more graph parameters and prove a number of new results related to implicit representation and factorial properties.

翻译：如何在计算机内存中高效地表示图是一个基础的数据结构问题。本文从组合角度探讨该问题。一个$n$顶点图$G$的表示称为隐式表示，若它为$G$的每个顶点分配一个长度为$O(\log n)$的二进制码，使得两顶点间的邻接关系是其码的函数。图的可遗传类$X$存在隐式表示的必要条件是$X$具有至多阶乘级增长速度。然而，如[Hatami & Hatami, FOCS 2022]最近所示，该条件并不充分。现有若干关于隐式表示存在性的充分条件涉及某些参数的有界性，如退化度或团宽。本文分析更多图参数，并证明一系列与隐式表示及阶乘性质相关的新结果。