• Corpus ID: 244708976

# The Implicit Graph Conjecture is False

@article{Hatami2021TheIG,
title={The Implicit Graph Conjecture is False},
author={Hamed Hatami and Pooya Hatami},
journal={ArXiv},
year={2021},
volume={abs/2111.13198}
}
• Published 25 November 2021
• Mathematics
• ArXiv
An efficient implicit representation of an n-vertex graph G in a family 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 log(n)) graphs on n vertices, and thus has at most factorial speed of growth. The Implicit Graph…
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## References

SHOWING 1-10 OF 30 REFERENCES
Implicit representations and factorial properties of graphs
• Mathematics, Computer Science
Discret. Math.
• 2015
Universal Graphs for Bounded-Degree Trees and Planar Graphs
• Mathematics
SIAM J. Discret. Math.
• 1989
It is shown that the minimum universal graph containing all bounded-degree graphs on n vertices with separators of size $n^\alpha$ has $O(n)$ edges if $\alpha \frac{1}{2}$.
Shorter Implicit Representation for Planar Graphs and Bounded Treewidth Graphs
• Computer Science, Mathematics
ESA
• 2007
A new implicit representation of planar graphs using asymptotically 2 log n-bit labels is proposed, and all the labels can be constructed in O(n log n) time, and support constant time adjacency testing.
Adjacency Labeling Schemes and Induced-Universal Graphs
• Mathematics, Computer Science
STOC
• 2015
An induced-universal graph for n-vertex graphs containing only O(2n/2) vertices is obtained, which is optimal up to a multiplicative constant, solving an open problem of Vizing from 1968.
Optimal Induced Universal Graphs and Adjacency Labeling for Trees
• Computer Science, Mathematics
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
• 2015
We show that there exists a graph G with O(n) nodes, where any forest of n nodes is a node-induced subgraph of G. Furthermore, for constant arboricity k, the result implies the existence of a graph
Asymptotically optimal induced universal graphs
We prove that the minimum number of vertices of a graph that contains every graph on k vertices as an induced subgraph is $${(1+o(1)) 2^{(k-1)/2}}$$(1+o(1))2(k-1)/2. This improves earlier estimates
Randomized Communication and the Implicit Graph Conjecture
• Mathematics
ArXiv
• 2021
It is conjecture that this always holds, i.e. that constant-cost randomized communication problems correspond to the set of stable families that satisfy the IGC, and also obtains constant-size adjacency sketches for Cartesian products and stable families of bounded twin-width.
Small induced-universal graphs and compact implicit graph representations
• Computer Science, Mathematics
The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
• 2002
We show that there exists a graph G with n /spl middot/ 2/sup O(log* n)/ nodes, where any forest with n nodes is a node-induced subgraph of G. Furthermore, the result implies the existence of a graph
Local representations using very short labels
Graphs which Contain all Small Graphs
• Mathematics
Eur. J. Comb.
• 1981