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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 114.40004

**Autor: ** Erdös, Pál

**Title: ** Remarks on a paper of Pósa (In English)

**Source: ** Publ. Math. Inst. Hung. Acad. Sci., Ser. A 7, 227-229 (1962).

**Review: ** Let G be a graph containing n vertices and k a natural number (1 \leq k < n/2). Denote by m_{k} the maximum of {n-k \choose 2}+k^{2} and {n-[(n-1)/2] \choose 2}+**[**{(n-1) \over 2} **]**^{2}.

Theorem: If each vertex of G has a degree \geq k and G contains m_{k}+1 edges, then G has a Hamilton line. The conclusion does not hold in general if G contains only m_{k} edges.

The final part of the paper deals with conditions which assure the existence of an open Hamilton line; the conditions in question are resulting from combining the conditions of the theorem of *L.Pósa* (reviewed above, Zbl 114.40003) and of the first theorem.

**Reviewer: ** A.Ádám

**Classif.: ** * 05C45 Eulerian and Hamiltonian graphs

**Index Words: ** topology

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