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.

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.

In 1978, C. Thomassen proved that in any graph one can destroy all the longest cycles by deleting at most one third of the vertices. We show that for graphs with circumference k≤8 it suffices to remove at most 1/k of the vertices. The Petersen graph demonstrates that this result cannot be extended to include k=9 but we show that in every graph with circumference nine we can destroy all 9-cycles by removing 1/5 of the vertices. We consider the analogous problem for digraphs and show that for digraphs with circumference k=2,3, it suffices to remove 1/k of the vertices. However this does not hold for k≥4.

A digraph is k-traceable if its order is at least k and each of its subdigraphs of order k is traceable. An oriented graph is a digraph without 2-cycles. The 2-traceable oriented graphs are exactly the nontrivial tournaments, so k-traceable oriented graphs may be regarded as generalized tournaments. It is well-known that all tournaments are traceable. We denote by t(k) the smallest integer bigger than or equal to k such that every k-traceable oriented graph of order at least t(k) is traceable. The Traceability Conjecture states that t(k) ≤ 2k-1 for every k ≥ 2 [van Aardt, Dunbar, Frick, Nielsen and Oellermann, A traceability conjecture for oriented graphs, Electron. J. Combin., 15(1):#R150, 2008]. We show that for k ≥ 2, every k-traceable oriented graph with independence number 2 and order at least 4k-12 is traceable. This is the last open case in giving an upper bound for t(k) that is linear in k.

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.

A digraph is $k$-traceable if its order is at least $k$ and each of its subdigraphs of order $k$ is traceable. The Traceability Conjecture (TC) states that for $k\geq 2$ every $k$-traceable oriented graph of order at least $2k-1$ is traceable. It has been shown that for $2\leq k\leq 6$, every $k$-traceable oriented graph is traceable. We develop an iterative procedure to extend previous results regarding the TC. In particular, we prove that every $7$-traceable oriented graph of order at least 9 is traceable and every 8-traceable graph of order at least 14 is traceable.

A (di)graph G is k-traceable (k >= 2) if every induced sub(di)graph of G that has order k is traceable. In particular, a 2-traceable graph is complete and a 2-traceable oriented graph is a tournament. In this paper we give structural characterizations of k-traceable graphs for each k <= 6. We also characterize, for k <= 4, those graphs that have a k-traceable orientation.

It is known that there exists a 2-connected claw-free maximal nontraceable graph of order n for every n >= 18. We demonstrate the existence of a 3-connected claw-free maximal nontraceable graph of order n for every n >= 60.

A digraph of order at least k is k-traceable if each of its subdigraphs of order k is traceable. We note that 2-traceable oriented graphs are tournaments and for k≥3, k-traceable oriented graphs can be regarded as generalized tournaments. We show that for 2≤k≤6 every k-traceable oriented graph is traceable, thus extending the well-known fact that every tournament is traceable. This result does not extend to k=7. In fact, for every k≥7, except possibly for k=8 or 10, there exist k-traceable oriented graphs that are nontraceable. However, we show that for every k≥2 there exists a smallest integer t(k) such that every k-traceable oriented graph of order at least t(k) is traceable.

The detour order of an oriented graph D, denoted by lambda(D), is the order of a longest path in D. An oriented graph is said to be k-detour saturated if lambda(D) <= k and lambda(D + xy) > k for any two non-adjacent vertices x and y in D. In this paper we characterize acyclic k-detour saturated oriented graphs. Since the strong component digraph of an oriented graph is acyclic, this characterization enables us to gain some insight into the structure of more complex oriented detour saturated graphs. We show that the maximum size of k-detour saturated oriented graphs of order n is the Turan number t (n, k), while the minimum size is O (n).

A (di)graph $G$ of order $n$ is $k$-traceable (for some $k$, $1\leq k\leq n$) if every induced sub(di)graph of $G$ of order $k$ is traceable. It follows from Dirac's degree condition for hamiltonicity that for $k\geq2$ every $k$-traceable graph of order at least $2k-1$ is hamiltonian. The same is true for strong oriented graphs when $k=2,3,4,$ but not when $k\geq5$. However, we conjecture that for $k\geq2$ every $k$-traceable oriented graph of order at least $2k-1$ is traceable. The truth of this conjecture would imply the truth of an important special case of the Path Partition Conjecture for Oriented Graphs. In this paper we show the conjecture is true for $k \leq 5$ and for certain classes of graphs. In addition we show that every strong $k$-traceable oriented graph of order at least $6k-20$ is traceable. We also characterize those graphs for which all walkable orientations are $k$-traceable.

We say that a function f:V→{0,1,…,diam(G)} is a broadcast if for every vertex v∈V, f(v)⩽e(v), where diam(G) denotes the diameter of G and e(v) denotes the eccentricity of v. The cost of a broadcast is the value f(V)=∑v∈Vf(v). In this paper we introduce and study the minimum and maximum costs of several types of broadcasts in graphs, including dominating, independent and efficient broadcasts.

The detour order of a graph G, denoted by τ(G), is the order of a longest path in G. The Path Partition Conjecture (PPC) is the following: If G is any graph and (a,b) any pair of positive integers such that τ(G)=a+b, then the vertex set of G has a partition (A,B) such that τ(〈A〉)⩽a and τ(〈B〉)⩽b. We prove that this conjecture is true for the class of claw-free graphs. We also show that to prove that the PPC is true, it is sufficient to consider the class of 2-connected graphs.

The vertex set and arc set of a digraph D are denoted by V (D) and E (D), respectively, and the number of vertices in a digraph D is denoted by n (D). A directed cycle (path, walk) in a digraph will simply be called a cycle (path, walk). A graph or digraph is called hamiltonian if it contains a cycle that visits every vertex, traceable if it contains a path that visits every vertex, and walkable if it contains a walk that visits every vertex. A digraph D is called strong (or strongly connected) if every vertex of D is reachable from every other vertex. Thus a digraph D of order bigger than 1 is strong if and only if it contains a closed walk that visits every vertex. A maximal strong subdigraph of a digraph D is called a strong component of D and a maximal walkable subdigraph of D is called a walkable component of D. A longest path in a digraph D is called a detour of D. The order of a detour of D is called the detour order of D and is denoted by λ (D) . If (a, b) is a pair of positive integers, a partition (A,B) of the vertex set of a digraph D is called an (a, b)-partition if λ(D〈A〉) ≤ a and λ(D〈B〉) ≤ b. If a digraph D has an (a, b)-partition for every pair of positive integers (a, b) such that a + b = λ (D), then D is called λ-partitionable. The Directed Path Partition Conjecture (DPPC) states:

A set S subset of V is a dominating set in a graph G = (V, E) if each vertex in V-S is adjacent to at least one vertex in S. The domination number gamma(G) is the minimum cardinality of a dominating set in G. We find improved bounds for gamma(G) + gamma(G) for graphs which have a given minimum degree.

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.

A graph is said to be detour‐saturated if the addition of any edge results in an increased greatest path length. In this paper, we add to the relatively small amount that is known about detour‐saturated graphs. Our main result is a determination of all connected detour‐saturated graphs with exactly one cycle. (The family of detour‐saturated trees was found by Kászonyi and Tuza 7 .) We also show that the smallest detour‐saturated graph of girth 5 is the graph obtainable from the Petersen graph by splitting one of its vertices into three, each of degree 1. © 2005 Wiley Periodicals, Inc. J Graph Theory

Let G be a graph with n vertices and let D be a minimum dominating set of G. If V - D contains a dominating set D' of G, then D' is called an inverse dominating set of G with respect to D. The inverse domination number gamma'(G) of G is the cardinality of a smallest inverse dominating set of G. In this paper we characterise graphs for which gamma(G) + gamma'(G) = n. We give a lower bound for the inverse domination number of a tree and give a constructive characterisation of those trees which achieve this lower bound.

The detour order τ(G) of a graph G is the order of a longest path of G. If S is a subset of V(G) such that the graph induced by S has detour order at most n, then S is called a Pn+1-free set in G. The Path Partition Conjecture (PPC) can be stated as follows: For any graph G and any positive integer n<τ(G),there exists a Pn+1-free set H in G such that τ(G-H)⩽τ(G)-n. We prove that if G is any graph and M is any maximal Pn+1-free set in G, then τ(G-M)⩽τ(G)-23(n+1). We also prove that if G has no cycle of order less than n or greater than τ(G)-n+2, then τ(G-M)⩽τ(G)-n for every maximal Pn+1-free subset M of G. As a corollary of the latter result we prove that the PPC is true for the class of connected, weakly pancyclic graphs.