The Square of Paths and Cycles

The square of a path (cycle) is the graph obtained by joining every pair of vertices of distance two in the path (cycle). Let G be a graph on n vertices with minimum degree δ(G). Posá conjectured that if δ(G) ≥ 23n, then G contains the square of a hamiltonian cycle. This is also a special case-of a conjecture of Seymour. In this paper, we prove that for any ϵ > 0, there exists a number m, depending only on ϵ, such that if δ(G) ≥ (23 + ϵ) n + m, then G contains the square of a hamitonian path between any two edges, which implies the squares of a hamiltonian cycle.