Cycles of many lengths in digraphs with Meyniel-like condition

Abstract

C. Thomassen (Proc. London Math. Soc. (3) 42 (1981), 231-251) gave a characterization of strongly connected non-Hamiltonian digraphs of order $p\ge 3$ with minimum degree $p-1$. In this paper we give an analogous characterization of strongly connected non-Hamiltonian digraphs with Meyniel-type condition (the sum of degrees of every pair of non-adjacent vertices $x$ and $y$ at least $2p-2$). Moreover, we prove that such digraphs $D$ contain cycles of all lengths $k$, for $2\le k\le m$, where $m$ is the length of a longest cycle in $D$.

19 pages, A detailed proof of a theorem published earlier by author in "Cycles of any length in digraphs with large semi-degrees", Academy Nauk Armyan SSR, Doklady 75(4) (1982) 147-152