Implicit Representations and Factorial Properties of Graphs

TL;DR

This paper explores implicit graph representations, introduces tools to identify classes with factorial growth, and demonstrates many such classes admit implicit representations, advancing understanding of graph encoding complexity.

Contribution

The paper introduces new tools to determine when hereditary graph classes have factorial growth and proves many such classes admit implicit representations, advancing the implicit graph conjecture.

Findings

Identified new hereditary classes with factorial growth.

Many classes with factorial growth admit implicit representations.

Provided tools to analyze the growth and representation of graph classes.

Abstract

The idea of implicit representation of graphs was introduced in [S. Kannan, M. Naor, S. Rudich, Implicit representation of graphs, SIAM J. Discrete Mathematics, 5 (1992) 596--603] and can be defined as follows. A representation of an $n$-vertex graph $G$ is said to be 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. Since an implicit representation of an $n$-vertex graph uses $O(n\log n)$ bits, any class of graphs admitting such a representation contains $2^{O(n\log n)}$ labelled graphs with $n$ vertices. In the terminology of [J. Balogh, B. Bollob\'{a}s, D. Weinreich, The speed of hereditary properties of graphs, J. Combin. Theory B 79 (2000) 131--156] such classes have at most factorial speed of growth. In this terminology, the implicit graph conjecture can be stated as follows: every class with at most factorial speed of growth which is hereditary admits an implicit representation. The question of deciding whether a given hereditary class has at most factorial speed of growth is far from being trivial. In the present paper, we introduce a number of tools simplifying this question. Some of them can be used to obtain a stronger conclusion on the existence of an implicit representation. We apply our tools to reveal new hereditary classes with the factorial speed of growth. For many of them we show the existence of an implicit representation.