Fleischner's theorem

Fleischner's theorem states that the square of every 2-connected graph contains a Hamiltonian cycle. The square joins distinct vertices whose distance in the original graph is at most two.

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Graph square: The square of a graph joins two distinct vertices when their distance in the graph is at most two. This graph operation is the structure whose Hamiltonicity the theorem guarantees.
Graph: A mathematical structure consisting of vertices and edges connecting pairs of vertices. Fleischner's theorem applies to graphs and constructs another graph from each one.
Herbert Fleischner: An Austrian mathematician known for work in graph theory, including the theorem bearing his name. He proved the result that every 2-connected graph has a Hamiltonian square.
Hamiltonian path: A path that visits every vertex of a graph exactly once. The theorem guarantees a closed spanning cycle in the square, not merely a spanning path.
2-vertex-connected graph: A graph with at least three vertices is 2-vertex-connected if it remains connected after deletion of any one vertex. This is the precise connectivity condition required by the theorem.
Walk (graph theory): A sequence of vertices in which consecutive vertices are joined by edges; vertices and edges may repeat. Short walks in the original graph explain adjacency in its square.
Graph power: A graph formed by joining vertices whose distance in the original graph is at most a specified integer. Fleischner's theorem is a foundational Hamiltonicity result about the second graph power.
Toughness (graph theory): A graph parameter measuring how many components can result from deleting a set of vertices relative to the set's size. Hamiltonicity research studies stronger structural conditions that can ensure spanning cycles.
Hamiltonian cycle: A cycle that visits every vertex of a graph exactly once before returning to its start. Fleischner's theorem guarantees such a cycle in the graph square.
Path (graph theory): A sequence of distinct vertices joined consecutively by edges. Paths of length at most two supply the connections used by the square.