Cycles in [formula omitted]-traceable oriented graphs

Abstract

A digraph of order at least k is termed k - traceable if each of its subdigraphs of order k is traceable. It turns out that several properties of tournaments—i.e., the 2-traceable oriented graphs—extend to k -traceable oriented graphs for small values of k . For instance, the authors together with O. Oellermann have recently shown that for k = 2 , 3 , 4 , 5 , 6 , all k -traceable oriented graphs are traceable. Moon [J.W. Moon, On subtournaments of a tournament, Canad. Math. Bull. 9(3) (1966) 297–301] observed that every nontrivial strong tournament T is vertex-pancyclic—i.e., through each vertex there is a cycle of every length from 3 up to the order of T . The present paper reports results pertaining to various cycle properties of strong k -traceable oriented graphs and explores the extent to which pancyclicity is retained by strong k -traceable oriented graphs. For each k ≥ 2 there are infinitely many k -traceable oriented graphs—e.g. tournaments. However, we establish an upper bound (linear in k ) on the order of k -traceable oriented graphs having a strong component with girth greater than 3. As an application of our findings, we show that the Path Partition Conjecture holds for 1-deficient oriented graphs having a strong component with girth at least 6. (A digraph is 1-deficient if its order is exactly one more than the order of its longest paths.)