Odd-Ramsey numbers of Hamilton cycles

TL;DR

This paper determines the asymptotic behavior of odd-Ramsey numbers for Hamilton cycles, showing they grow proportionally to the square root of the number of vertices, using finite-field constructions and combinatorial methods.

Contribution

It establishes the order of the odd-Ramsey number for Hamilton cycles and introduces new bounds and techniques, including finite-field constructions and parity switch frameworks.

Findings

r_odd(n,C_n) = Θ(√n) for Hamilton cycles

Finite-field construction provides upper bounds

Parity switch methods establish lower bounds

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 $r_{{\text odd}}(n,C_n) = \Theta(\sqrt{n})$. 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.