Asian Research Journal of Mathematics

A longstanding question in extremal combinatorics asks for the edge threshold in n-vertex graphs that guarantees the existence of a cycle in which each vertex is connected to its antipodal counterpart. The problem addressed in this article concerns cycles where each vertex maintains a connection to its antipodal counterpart, creating a rich geometric structure that has implications for both theoretical understanding and practical applications. The study establishes that \(Ω(n^{3⁄2})\) edges suffice, with bounds tight up to constant factors. The proof introduces a new lemma for locating near-spanning expanders in auxiliary C₄ -graphs and leverages a bipartite construction sensitive to parity in diagonal cycle configurations. This expansion-driven method overcomes limitations of traditional bipartite techniques and advances tools in extremal graph theory.