Tight bounds on adjacency labels for monotone graph classes

TL;DR

The paper delivers a nearly complete characterization of adjacency labeling schemes for monotone graph classes across a broad growth spectrum. It introduces the notions of $f$-good graphs and decent functions to construct fast-growing monotone classes with provable lower bounds, and simultaneously shows a matching $O(f(n)\log n)$ upper bound on label sizes for any class of speed $2^{O(nf(n))}$, leaving a gap of a factor $\Theta(\log n)$ from the information-theoretic lower bound $\Omega(f(n))$. A key positive result is that monotone small classes have bounded degeneracy and thus admit information-theoretically optimal implicit representations, including all monotone small classes, while factorial monotone classes exhibit refuted implicit representations in general. The work also clarifies the relationship between degeneracy, implicit representations, and the existence of universal graphs, and raises important open questions about hereditary small classes and the full scope of the Implicit Graph Conjecture within monotone and small families.

Abstract

A class of graphs admits an adjacency labeling scheme of size $b(n)$, if the vertices in each of its $n$-vertex graphs can be assigned binary strings (called labels) of length $b(n)$ so that the adjacency of two vertices can be determined solely from their labels. We give tight bounds on the size of adjacency labels for every family of monotone (i.e., subgraph-closed) classes with a well-behaved growth function between $2^{O(n \log n)}$ and $2^{O(n^{2-δ})}$ for any $δ> 0$. Specifically, we show that for any function $f: \mathbb N \to \mathbb R$ satisfying $\log n \leqslant f(n) \leqslant n^{1-δ}$ for any fixed $δ> 0$, and some~sub-multiplicativity condition, there are monotone graph classes with growth $2^{O(nf(n))}$ that do not admit adjacency labels of size at most $f(n) \log n$. On the other hand, any such class does admit adjacency labels of size $O(f(n)\log n)$. Surprisingly this tight bound is a $Θ(\log n)$ factor away from the information-theoretic bound of $Ω(f(n))$. The special case when $f = \log$ implies that the recently-refuted Implicit Graph Conjecture [Hatami and Hatami, FOCS 2022] also fails within monotone classes. We further show that the Implicit Graph Conjecture holds for all monotone \emph{small} classes. In other words, any monotone class with growth rate at most $n!\,c^n$ for some constant $c>0$, admits adjacency labels of information-theoretic order optimal size. In fact, we show a more general result that is of independent interest: any monotone small class of graphs has bounded degeneracy.We conjecture that the Implicit Graph Conjecture holds for all hereditary small classes.

Figures (1)

• Figure 1: A $\checkmark$ indicates that all classes of the given type have an implicit representation, a ✗ shows that they do not, and a ${ \emph{?}}$ signals that the question is open. A $\checkmark$ is inherited by every sub-region, a ✗ is inherited to the left of the marked region, and a ${ \emph{?}}$ only holds in that region. The upper and lower bounds (UB and LB respectively) are stated up to constants which may depend on the class. The dashed extension of the bounded degeneracy region illustrates its containment of monotone small classes (\ref{['cor:monSmall-ir']}).

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Conjecture 1.7: Small Implicit Graph Conjecture
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3