The Implicit Graph Conjecture is False

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 $n^{\Omega(1)}$.

Authors

• Hamed Hatami
• Pooya Hatami

Keywords

• Computer science
• Binary codes
• Labeling
• Conjecture
• Pair Of Vertices
• Efficient Representation
• Implicit Representation
• Family Of Graphs
• High Probability
• Complex Communication
• Representative Families
• Random Graph
• Set Of Graphs
• Graph Size
• Induced Subgraph
• Number Of Graphs
• Decoding Function
• Universal Representation
• Implicit Graph Conjecture
• Hereditary Graph Family
• Universal Graph
• Adjacency Labeling Scheme