On the clique number of random Cayley graphs and related topics

2024-12-30

AI Summary

This work investigates the clique number (size of the largest complete subgraph) and its dual, the independence number, of random Cayley graphs. For random Cayley graphs on abelian groups of order $N$, we develop a purely combinatorial, group-structure-agnostic approach—bypassing representation-theoretic machinery—to significantly improve the classical $O(log^2 N)$ upper bound on the clique number to $O(log N cdot log log N)$, nearly matching the Ramsey-theoretic lower bound. This confirms Alon’s conjecture that both clique and independence numbers are $O(log N)$ for almost all abelian groups. Furthermore, on the vector space $mathbb{F}_q^n$ with characteristic congruent to $1 pmod{4}$, we construct a family of self-complementary Cayley graphs whose clique and independence numbers are at most $(2+o(1))log N$, achieving asymptotic optimality. The results integrate extremal combinatorics, probabilistic methods, and representation-theoretic insights, advancing the frontier of random algebraic graph theory.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We prove that a random Cayley graph on a group of order $N$ has clique number $O(log N log log N)$ with high probability. This bound is best possible up to the constant factor for certain groups, including~$mathbb{F}_2^n$, and improves the longstanding upper bound of $O(log^2 N)$ due to Alon. Our proof does not make use of the underlying group structure and is purely combinatorial, with the key result being an essentially best possible upper bound for the number of subsets of given order that contain at most a given number of colors in a properly edge-colored complete graph. As a further application of this result, we study a conjecture of Alon stating that every group of order $N$ has a Cayley graph whose clique number and independence number are both $O(log N)$, proving the conjecture for all abelian groups of order $N$ for almost all $N$. For finite vector spaces of order $N$ with characteristic congruent to $1 pmod 4$, we prove the existence of a self-complementary Cayley graph on the vector space whose clique number and independence number are both at most $(2+o(1))log N$. This matches the lower bound for Ramsey numbers coming from random graphs and solves, in a strong form, a problem of Alon and Orlitsky motivated by information theory.

Problem

• Random Cayley Graphs
• Maximum Clique Size
• Independent Set Size

Innovation

• Cayley Graphs
• Maximal Complete Subgraph
• Random Graph Theory