The Square of Paths and Cycles

Abstract 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 ). Posa conjectured that if δ( G ) ≥ 2 3 n , 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 ) ≥ ( 2 3 + ϵ) n + m , then G contains the square of a hamitonian path between any two edges, which implies the squares of a hamiltonian cycle.