Oriented Hamiltonian Cycles in Tournaments: a Proof of Rosenfeld's Conjecture

TL;DR

This paper proves Rosenfeld's conjecture that large enough tournaments contain all non-directed cycles of certain lengths, with only 35 exceptions, advancing understanding of cycle structures in directed graphs.

Contribution

The paper provides a proof that nearly all large tournaments contain every non-directed cycle of specified lengths, confirming Rosenfeld's conjecture with minimal exceptions.

Findings

Almost all large tournaments contain all non-directed cycles of lengths between 3 and n+1.

Only 35 specific tournaments are exceptions to the cycle containment.

The result confirms a long-standing conjecture in tournament graph theory.

Abstract

Rosenfeld in 1974 conjectured that there is an integer N > 8 such that every tournament of order n > N contains every non-directed cycle of order n. We prove that, with exactly 35 exceptions, every tournament of order n > 2 contains each non-directed cycle of order m, 2 < m < n+1.