Susan A. van Aardt

The directed path partition conjecture

The Directed Path Partition Conjecture is the following: If D is a digraph that contains no path with more than λ vertices then, for every pair (a, b) of positive integers with λ = a + b, there exists a vertex partition (A, B) of D such that no path in D〈A〉 has more than a vertices and no path in D〈B〉 has more than b vertices.We develop methods for finding the desired partitions for various classes of digraphs.

An Infinite Family of Planar Hypohamiltonian Oriented Graphs

Carsten Thomassen asked in 1976 whether there exists a planar hypohamiltonian oriented graph. We answer his question by presenting an infinite family of planar hypohamiltonian oriented graphs, the smallest of which has order 9. A computer search showed that 9 is the smallest possible order of a hypohamiltonian oriented graph.

Cycles in k-traceable oriented graphs

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.)

Hamiltonicity of locally hamiltonian and locally traceable graphs

Abstract If P is a given graph property, we say that a graph G is locally P if 〈 N ( v ) 〉 has property P for every v ∈ V ( G ) where 〈 N ( v ) 〉 is the induced graph on the open neighbourhood of the vertex v . We consider the complexity of the Hamilton Cycle Problem for locally traceable and locally hamiltonian graphs with small maximum degree. The problem is fully solved for locally traceable graphs with maximum degree 5 and also for locally hamiltonian graphs with maximum degree 6 (van Aardt et al., 2016). We show that the Hamilton Cycle Problem is NP-complete for locally traceable graphs with maximum degree 6 and for locally hamiltonian graphs with maximum degree 10. We also show that there exist regular connected nonhamiltonian locally hamiltonian graphs with connectivity 3, thus answering two questions posed by Pareek and Skupien (1983).

On Saito’s Conjecture and the Oberly–Sumner Conjectures

For a given graph property

Hypohamiltonian Oriented Graphs of All Possible Orders

\mathcal {P}

Every 8-traceable oriented graph is traceable

P, we say a graph G is locally