The detour order of a graph G, denoted by tau(G), is the order of a longest path in G. If a and b are positive integers and the vertex set of G can be partitioned into two subsets A and B such that tau((A)) <= a and tau((B)) <= b, we say that (A, B) is an (a, b)-partition of G. If equality holds in both instances, i.e., if tau((A)) = a and tau((B)) = b, we call (A, B) an exact (a, b)-partition. The Path Partition Conjecture asserts that if G is any graph and a, b any pair of positive integers such that tau(G) = a + b, then G has an (a, b)-partition. The Strong Path Partition Conjecture asserts that, under the same conditions, G has an exact (a, b)-partition. The Path Partition Conjecture is now more than 40 years old. It first appeared in the literature in a paper by Laborde, Payan and Xuong (1982). It is known that the Path Partition Conjecture holds for all a <= 8. The case a <= 5 was first proved by Vronka (1986), the case a = 6 by Dunbar and Frick (1999) and the cases a = 7 and a = 8 by Melnikov and Petrenko (2002 and 2005). Using a new partition strategy involving a recursive procedure, De Wet, Dunbar, Frick and Oellermann (2024) improved these results by showing that the Strong Path Partition Conjecture holds for a < 8. By expanding and refining the recursive procedure, we prove that the Strong Partition Conjecture also holds for a = 9.

The detour order of a graph G, denoted by tau (G), is the order of a longest path in G. If a and b are positive integers and the vertex set of G can be partitioned into two subsets A and B such that tau (A) <= a and tau (B) <= b, we say that (A, B) is an (a, b)-partition of G. If equality holds in both instances, we call (A, B) an exact (a, b)-partition. The Path Partition Conjecture (PPC) asserts that if G is any graph and a, b any pair of positive integers such that tau (G) = a + b, then G has an (a, b)-partition. The Strong PPC asserts that under the same circumstances G has an exact (a, b)-partition. While a substantial body of work in support of the PPC has been developed over the past three decades, no results on the Strong PPC have yet appeared in the literature. In this paper we prove that the Strong PPC holds for a <= 8.

The Path Partition Conjecture (PPC) is now at least forty years old. It states that if a and b are any positiveintegers and G is any graph that has no path with more than a + b vertices, then G has a vertex partition (A, B) such that the graph induced by A has no path with more than a vertices and the graph induced by B has no path with more than b vertices. It is known that the PPC holds for all a ≤ 8. The case a = 5 follows from a result proved by Vronka in 1986. In 1999, Dunbar and Frick proved the case a = 6 by finding the necessary partitions by means of a recursive procedure, which was extended by Melnikov and Petrenko to prove the case a = 7 in 2002 and the case a = 8 in 2005. Recently, de Wet, Dunbar, Frick and Oellermann developed a different recursive procedure, which yielded shorter proofs for all a ≤ 8. We extend this recursive procedure to prove that the PPC holds for a = 9.

In 1978 Thomassen asked whether planar hypohamiltonian oriented graphs exist. Infinite families of such graphs have since been described but for infinitely many n it remained an open question whether planar hypohamiltonian oriented graphs of order n exist. In this paper we develop new methods for constructing hypohamiltonian digraphs, which, combined with efficient graph generation algorithms, enable us to fully characterise the orders for which planar hypohamiltonian oriented graphs exist. Our novel methods also led us to discover the planar hypohamiltonian oriented graph of smallest order and size, as well as infinitely many hypohamiltonian orientations of maximal planar graphs. Furthermore, we answer a question related to a problem of Schiermeyer on vertex degrees in hypohamiltonian oriented graphs, and characterise all the orders for which planar hypotraceable oriented graphs exist.

A graph G is locally P, abbreviated LP, if for every vertex v in G the open neighbourhood N(v) of v is non-empty and induces a graph with property P. Specifically, a graph G without isolated vertices is locally connected (LC) if N(v) induces a connected graph for each v is an element of V (G), and locally hamiltonian (LH) if N(v) induces a hamiltonian graph for each v is an element of V (G). A graph G is locally locally P (abbreviated (LP)-P-2) if N(v) is non-empty and induces a locally P graph for every v is an element of V (G). This concept is generalized to an arbitrary degree of nesting. For any k 0 we call a graph locally k-nested-hamiltonian if it is (LC)-C-m for m = 0, 1, . . ., k and (LH)-H-k (with (LC)-C-0 and (LH)-H-0 meaning connected and hamiltonian, respectively). The class of locally k-nested-hamiltonian graphs contains important subclasses. For example, Skupien had already observed in 1963 that the class of connected LH graphs (which is the class of locally 1-nested-hamiltonian graphs) contains all triangulations of closed surfaces. We show that for any k >= 1 the class of locally k-nested-hamiltonian graphs contains all simple-clique (k + 2)-trees. In 1979 Oberly and Sumner proved that every connected K-1,K-3-free graph that is locally connected is hamiltonian. They conjectured that for k >= 1, every connected K-1,(k)+3-free graph that is locally (k + 1)-connected is hamiltonian. We show that locally k-nested-hamiltonian graphs are locally (k + 1)-connected and consider the weaker conjecture that every K-1,(k)+3-free graph that is locally k-nested-hamiltonian is hamiltonian. We show that if our conjecture is true, it would be "best possible" in the sense that for every k >= 1 there exist K-1,(k)+4-free locally k-nested-hamiltonian graphs that are non-hamiltonian. We also attempt to determine the minimum order of non-hamiltonian locally k-nested-hamiltonian graphs and investigate the complexity of the Hamilton Cycle Problem for locally k-nested-hamiltonian graphs with restricted maximum degree.

We say a graph G is locally P if for each vertex v in G the open neighbourholod of v induces a graph with property P. The Hamilton Cycle Problem (HCP) is the problem of deciding whether a graph contains a Hamilton cycle. It is known that the HCP is NP-complete for locally traceable (LT) graphs with maximum degree 6. We extend that result to 1-tough graphs and to graphs with restricted degree sequences. If R is a set of nonnegative integers, we say a graph G is R.-regular if the degrees of all the vertices in V(G) are elements of R. We show that the HCP is NP-complete for R-regular LT graphs if R is any set of natural numbers with max(R) >= 6, with the possible exception of {4, 6} and {6}. It is known that the HCP is NP-complete for LH graphs with maximum degree 10. We improve this result by showing that the HCP is NP-complete for 1-tough LH graphs with maximum degree 9 and for R-regular LH graphs if R. is any set of natural numbers with min(R) = 3 and max(R) is an element of {9, 10}, or if R. is any set of natural numbers with max(R) >= 11. Finally, we show that the HCP for k-connected LH graphs that are also locally (k - 1)-connected is NP-complete for every k >= 3. (C) 2019 Elsevier B.V. All rights reserved.

A graph G is locally connected if for every v∈V(G) the open neighbourhood N(v) of v is nonempty and induces a connected graph in G. We characterize locally connected graphs of order n with less than 2n edges and show that for any natural number k the Hamilton Cycle Problem for locally connected graphs of order n with m edges is polynomially solvable if m≤2n+klog2n, but NP-complete if m=2n+⌊n1∕k⌋.

If P is a given graph property, we say that a graph G is locally P if < N(v)> has property for every v is an element of 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). (C) 2017 Elsevier B.V. All rights reserved.

For a given graph property \(\mathcal {P}\), we say a graph G is locally \(\mathcal {P}\) if for each \(v \in V(G)\), the subgraph induced by the open neighbourhood of v has property \(\mathcal P\). A closed locally \(\mathcal {P}\) graph is defined analogously in terms of closed neighbourhoods. It is known that connected locally hamiltonian graphs are not necessarily hamiltonian. Saito (in Computational Geometry and Graph Theory, Lecture Notes in Computer Science, vol. 4535, pp. 191–200. Springer, Berlin, 2008) conjectured that if G is a graph of order at least 3 such that for every vertex v in G the subgraph induced by the closed neighbourhood N[v] of v satisfies the Chvátal–Erdős condition for hamiltonicity, then G is hamiltonian. Oberly and Sumner (in J Graph Theory 3:351–356, 1979) conjectured that if G is a connected, locally k-connected \(K_{1,k+2}\)-free graph of order at at least 3, then G is hamiltonian. We prove a result that lends support to both these conjectures. We also provide a framework for investigating these and other related conjectures.

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 Skupień (1983).

Local properties and their global implications date back to the 1960’s, when Skupien [23, 24] initiated the study of locally hamiltonian graphs. A graph is (globally) hamiltonian if it has a Hamilton cycle, i.e., a cycle that visits every vertex. Skupien called a graph G locally hamiltonian if for each vertex v in G, the subgraph induced by the open neighbourhood N(v) of v is hamiltonian. If X ⊆ V (G), we denote by 〈X〉 the subgraph of G induced by X. For a given graph property P, we call a graph G locally P if 〈N(v)〉 has property P for every v ∈ V (G). Locally traceable graphs were considered by Pareek and Skupien [21], and Chartrand and Pippert [8] introduced the study of locally connected graphs. For undefined concepts and notation we refer the reader to [6]. Global cycle properties of locally connected, locally traceable and locally hamiltonian graphs were studied, for example, in [1, 8, 13, 16, 20]. Another local property that has been studied in combination with local connectivity is the property of being claw-free, i.e., not having the claw, K1,3, as induced subgraph. Note that a graph G is claw-free if and only if α(〈N(v)〉) ≤ 2 for every v ∈ V (G) (where α denotes the vertex independence number). The following theorem of Oberly and Sumner [18] has sparked considerable interest in finding combinations of local properties that imply hamiltonicity.