The Implicit Graph Conjecture is False

TL;DR

This paper disproves the Implicit Graph Conjecture by showing that some hereditary graph families with factorial growth cannot be represented efficiently with codes of logarithmic length, challenging previous assumptions.

Contribution

It establishes the existence of hereditary graph families with factorial growth that require longer codes, refuting the conjecture that all such families admit efficient implicit representations.

Findings

Hereditary graph families with factorial growth can require codes of polynomial length.

The Implicit Graph Conjecture is false for certain hereditary graph families.

Some graph families defy efficient implicit representation despite their growth rate.

Abstract

An efficient implicit representation of an $n$-vertex graph $G$ in a family $\mathcal{F}$ of graphs assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency between every pair of vertices can be determined only as a function of their codes. This function can depend on the family but not on the individual graph. Every family of graphs admitting such a representation contains at most $2^{O(n\log(n))}$ graphs on $n$ vertices, and thus has at most factorial speed of growth. The Implicit Graph Conjecture states that, conversely, every hereditary graph family with at most factorial speed of growth admits an efficient implicit representation. We refute this conjecture by establishing the existence of hereditary graph families with factorial speed of growth that require codes of length $n^{\Omega(1)}$.