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} }
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…
3 Citations
Implicit representation of sparse hereditary families
- MathematicsArXiv
- 2022
It is proved that for every ε > 0 there is an integer d ≥ 1 so that if F is a hereditary family with speed f(n) ≤ 2(1/4−ε)n2 then Fn admits an implicit representation of size O(n logn).
A counter-example to the probabilistic universal graph conjecture via randomized communication complexity
- MathematicsArXiv
- 2021
We refute the Probabilistic Universal Graph Conjecture of Harms, Wild, and Zamaraev, which states that a hereditary graph property admits a constant-size probabilistic universal graph if and only if…
Sketching Distances in Monotone Graph Classes
- Computer Science
- 2022
It is shown that, for monotone classes of graphs, there is a strict hierarchy: approximate distance threshold sketches imply small-distance sketches, which imply adjacency sketches, whereas the reverse implications are each false.
References
SHOWING 1-10 OF 30 REFERENCES
Implicit representations and factorial properties of graphs
- Mathematics, Computer ScienceDiscret. Math.
- 2015
Universal Graphs for Bounded-Degree Trees and Planar Graphs
- MathematicsSIAM 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, MathematicsESA
- 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 ScienceSTOC
- 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, Mathematics2015 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
- Mathematics
- 2017
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
- MathematicsArXiv
- 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, MathematicsThe 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…