Computational Results on the Traceability of Oriented Graphs of Small Order

Abstract

A digraph $D$ is traceable if it contains a path visiting every vertex, and hypotraceable if $D$ is not traceable, but $D-v$ is traceable for every vertex $v\in V(D)$. Van Aardt, Frick, Katrenič and Nielsen [Discrete Math. 11(2011), 1273-1280] showed that there exists a hypotraceable oriented graph of order $n$ for every $n\geq 8$, except possibly for $n=9$ or $11$. These two outstanding existence questions for hypotraceable oriented graphs are settled in this paper --- the first in the negative and the second in the affirmative.  Furthermore, $D$ is $k$-traceable if $D$ has at least $k$ vertices and each of its induced subdigraphs of order $k$ is traceable.  It is known that for $k\leq 6$ every k-traceable oriented graph is traceable and that for $k=7$ and each $k\geq 9$ there exist nontraceable $k$-traceable oriented graphs of order $k+1$. The Traceability Conjecture states that for $k\geq 2$ every $k$-traceable oriented graph of order $n\geq 2k-1$ is traceable. In this paper it is shown via computer searches that all $7$-traceable and $8$-traceable oriented graphs of orders $9$, $10$ and $11$ are traceable, and that all $9$-traceable oriented graphs of order $11$ are traceable. All hypotraceable graphs of order $10$ are also found. Recently, these results are used to prove that the Traceability Conjecture also holds for $k =7, 8$ and 9, except possibly when $k=9$ and $22\leq n\leq 32$.