Pósa's theorem explained

Pósa's theorem, in graph theory, is a sufficient condition for the existence of a Hamiltonian cycle based on the degrees of the vertices in an undirected graph. It implies two other degree-based sufficient conditions, Dirac's theorem on Hamiltonian cycles and Ore's theorem. Unlike those conditions, it can be applied to graphs with a small number of low-degree vertices. It is named after Lajos Pósa, a protégé of Paul Erdős born in 1947, who discovered this theorem in 1962.

The Pósa condition for a finite undirected graph

G having n vertices requires that, if the degrees of the n vertices in increasing order asd _{1} \leq d _{2} \leq ... \leq d _{n} , then for each index k < n/ 2 the inequality k < d _{k} is satisfied.

Pósa's theorem states that if a finite undirected graph satisfies the Pósa condition, then that graph has a Hamiltonian cycle in it.

References

• • Katona–Recski–Szabó: A számítástudomány alapjai, Typotex, Budapest, 2003, (Hungarian undergraduate level course book).

External links

• • About the Pósa theorem