Exact values and improved bounds on the clique number of cyclotomic graphs
Chi Hoi Yip
TL;DR
The paper addresses the problem of bounding the clique number of cyclotomic graphs, including generalized Paley graphs, over non-square field orders. It introduces a refined Stepanov-style approach, leveraging Rédei polynomials and a direction-set bound to convert additive-structure information into clique-number estimates. The main contributions are a new upper bound $\omega(\operatorname{Cay}(\mathbb{F}_q^+;S)) \le \sqrt{|S/S|}+\sqrt{q/p}$ for odd $q$, asymptotically sharp results for an infinite family of generalized Paley graphs, and exact determinations of clique numbers for several infinite GP(q,d) families (e.g., $q=p^3$, $d|p^2+p+1$, $d>p$ gives $\omega=p$). Additionally, the paper proves a strengthened lower bound on the number of directions determined by large Cartesian products in $AG(2,q)$, with sharp instances. These results broaden nontrivial clique-number bounds to non-square orders and connect arithmetic combinatorics with finite geometry, yielding both asymptotic and exact outcomes in structured graph families.
Abstract
Let $q$ be an odd power of a prime $p$, and $S \subset \mathbb{F}_q^*$ such that $S=-S$ and $S/S \neq \mathbb{F}_q^*$. We show that the clique number of the Cayley graph $\operatorname{Cay}(\mathbb{F}_q^+,S)$ is at most $\sqrt{|S/S|}+\sqrt{q/p}$, improving the best-known $\sqrt{q}$ upper bound for many families of such graphs substantially. Such a new bound is strongest for cyclotomic graphs and in particular, it implies the first nontrivial upper bound on the clique number of all generalized Paley graphs of non-square order, extending the work of Hanson and Pertidis. Moreover, our new bound is asymptotically sharp for an infinite family of generalized Paley graphs, and we further discover the first nontrivial family among them for which the clique number can be exactly determined. We also obtain a new lower bound on the number of directions determined by a large Cartesian product in the affine Galois plane $AG(2,q)$, which is sharp for infinite families.