[Paper Review] Explicit tough Ramsey graphs

Lê Anh Vinh, Dang Phuong Dung|ArXiv.org|Jul 17, 2008

Limits and Structures in Graph Theory5 references5 citations

TL;DR

This paper constructs explicit triangle-free graphs using finite Euclidean and non-Euclidean planes over finite fields, demonstrating that for infinitely many prime powers q = 12k+5, these graphs are t-tough yet not pancyclic, thus providing new counterexamples to Chvátal's toughness conjecture. The graphs also yield a constructive lower bound of R(3,k) ≥ Ω(k^{4/3}) for the Ramsey number R(3,k).

ABSTRACT

A graph G is t-tough if any induced subgraph of it with x > 1 connected components is obtained from G by deleting at least tx vertices. Chvatal conjectured that there exists an absolute constant t_0 so that every t_0-tough graph is pancyclic. This conjecture was disproved by Bauer, van den Heuvel and Schmeichel by constructing a t_0-tough triangle-free graph for every real t_0. For each finite field F_q with q odd, we consider graphs associated to the finite Euclidean plane and the finite upper half plane over F_q. These graphs have received serious attention as they have been shown to be Ramanujan (or asymptotically Ramanujan) for large q. We will show that for infinitely many q, these graphs provide further counterexamples to Chvatal's conjecture. They also provide a good constructive lower bound for the Ramsey number R(3,k).

Motivation & Objective

• To disprove Chvátal's conjecture that every t₀-tough graph is pancyclic by constructing explicit counterexamples.
• To provide explicit, combinatorially defined graphs that are triangle-free and highly tough, using finite field constructions.
• To establish a constructive lower bound for the Ramsey number R(3,k) using the chromatic number and independence number of finite geometry-based graphs.
• To analyze the toughness, girth, diameter, and chromatic number of finite quadrance and non-Euclidean graphs over F_q.
• To explore the limits of known chromatic number bounds for triangle-free regular graphs via finite field varieties.

Proposed method

• Construct finite Euclidean graphs D_q(a) using quadrance Q(X,Y) = (y₁−x₁)² + (y₂−x₂)² over F_q², with edges when quadrance equals a fixed a ∈ F_q.
• Construct finite non-Euclidean graphs V_q(σ,a) on the finite Poincaré upper half-plane H_q = {x + y√σ | x,y ∈ F_q, y ≠ 0}, with edges when Poincaré distance d(z,w) = a.
• Use algebraic geometry over finite fields to prove that for q = 12k+5, the graph V_q(3,6) is triangle-free and has high chromatic number.
• Apply bounds on independence number via character sum estimates and exponential sum techniques to derive lower bounds on chromatic number.
• Establish that the chromatic number of V_q(3,6) exceeds (0.5 + o(1))n_q^{1/4}, where n_q = q² − q.
• Derive a lower bound for the Ramsey number R(3,k) by relating the chromatic number and independence number of the constructed graphs.

Experimental results

Research questions

• RQ1 Do there exist explicit, triangle-free, t-tough graphs for arbitrarily large t that are not pancyclic, thus refuting Chvátal's toughness conjecture?
• RQ2 Can finite geometry over F_q provide explicit constructions of Ramsey graphs with strong lower bounds on R(3,k)?
• RQ3 What is the chromatic number of finite non-Euclidean graphs V_q(σ,a) over F_q when q = 12k+5?
• RQ4 How do the independence number and chromatic number of quadrance and non-Euclidean graphs behave asymptotically as q → ∞?
• RQ5 Can higher-dimensional finite field varieties yield better bounds for the chromatic number of triangle-free regular graphs?

Key findings

• For infinitely many q = 12k+5, the finite non-Euclidean graph V_q(3,6) is triangle-free and t-tough for arbitrarily large t, disproving Chvátal’s conjecture.
• The chromatic number of V_q(3,6) satisfies χ(V_q(3,6)) ≥ (0.5 + o(1))n_q^{1/4}, where n_q = q² − q.
• The independence number of V_q(3,6) is at most (2 + o(1))n_q^{3/4}, leading to the chromatic number lower bound.
• The construction yields a constructive lower bound R(3,k) ≥ Ω(k^{4/3}) for the Ramsey number R(3,k).
• The finite Euclidean graph D_q(a) is (q+1)-regular and triangle-free, with chromatic number bounded by O(q / log₂q).
• The graphs D_q(a) and V_q(σ,a) are Ramanujan or asymptotically Ramanujan for large q, ensuring strong spectral properties.