Odd-Ramsey numbers of Hamilton cycles
Simona Boyadzhiyska, Shagnik Das, Thomas Lesgourgues, Kalina Petrova
TL;DR
This work studies the odd-Ramsey number $r_{\text{odd}}(n,H)$ with $H=C_n$, determining $r_{\text{odd}}(n,C_n)$ up to a constant factor in the complete-graph setting: $\big(\frac{\sqrt{2}}{2}+o(1)\big)\sqrt{n} \le r_{\text{odd}}(n,C_n) \le \frac{3\sqrt{2}}{2}\sqrt{n}$. The upper bound uses an explicit finite-field construction, while the lower bound relies on a parity-switch framework built around switches and a SPICy (Switches Placed in Cycles) structure, culminating in a Hamilton cycle with at most one odd colour class. The paper also initiates the study of odd-Ramsey numbers for Hamilton cycles in Dirac graphs, proving that a small increase above the Dirac threshold $n/2$ forces nontrivial odd-Ramsey numbers, and providing bounds for $r_{\text{odd}}(\ast,n,\delta;C_n)$ that depend on the minimum degree. Overall, the authors blend algebraic constructions, parity-switching arguments, and absorption techniques to map the complete-graph landscape and begin the sparse-graph regime, opening several natural directions for further refinement and related Hamiltonian-structural questions.
Abstract
The odd-Ramsey number $r_{\text odd}(n,H)$ of a graph $H$, as introduced by Alon in his work on graph-codes, is the minimum number of colours needed to edge-colour $K_n$ so that every copy of $H$ intersects some colour class in an odd number of edges. In this paper, we determine the odd-Ramsey number of Hamilton cycles up to a small multiplicative factor, proving that $r_{\text odd}(n,C_n) = Θ(\sqrt{n})$. Our upper bound follows from an explicit finite-field construction, while the matching lower bound uses a combinatorial framework based on parity switches. We also initiate the study of odd-Ramsey numbers of Hamilton cycles in Dirac graphs, demonstrating that a small increase in the minimum degree beyond $n/2$ forces nontrivial odd-Ramsey numbers.