Cycles of many lengths in digraphs with Meyniel-like condition

TL;DR

This paper characterizes strongly connected non-Hamiltonian digraphs satisfying a Meyniel-type degree condition and proves they contain cycles of all lengths up to their longest cycle.

Contribution

It provides an analogous characterization to Thomassen's for digraphs with Meyniel-like conditions and establishes the existence of cycles of all lengths up to the maximum cycle length.

Findings

Characterization of strongly connected non-Hamiltonian digraphs with Meyniel-type condition

Existence of cycles of all lengths up to the longest cycle in such digraphs

Extension of Thomassen's characterization to Meyniel-like conditions

Abstract

C. Thomassen (Proc. London Math. Soc. (3) 42 (1981), 231-251) gave a characterization of strongly connected non-Hamiltonian digraphs of order $p\geq 3$ 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 $2\leq k\leq m$, where $m$ is the length of a longest cycle in $D$.