On Hamiltonian Powers of Digraphs

 For a strongly connected digraph D, the k-th power D ^k of D is the digraph with the same set of vertices, a vertex x being joined to a vertex y in D ^k if the directed distance from x to y in D is less than or equal to k. It follows from a result of Ghouila-Houri that for every digraph D on n vertices and for every k≥n/2, D ^k is hamiltonian. In the paper we characterize these digraphs D of odd order whose (⌈n/2 ⌉−1)-th power is hamiltonian. Revised: June 13, 1997     

Randomly antitraceable digraphs

An antipath in a digraph is a semipath containing no (directed) path of length 2. A digraph D is randomly antitraceable if for each vertex v of D, any antipath beginning at v can be extended to a hamiltonian antipath beginning at v. In this paper randomly antitraceable digraphs are characterized.     

c-Pancyclic Partial Ordering and （c-1）-Pan-Outpath Partial Ordering in Semicomplete Multipartite Digraphs

An outpath of a vertex v in a digraph is a path starting at v such that v dominates the end vertex of the path only if the end vertex also dominates v.First we show that letting D be a strongly connected semicomplete c-partite digraph (c≥3)1 and one of the partite sets of it consists of a single vertex, say v, then D has a c-pancyclic partial ordering from v, which generalizes a result about pancyclicity of multipartite tournaments obtained by Gutin in 1993.Then we prove that letting D be a strongly connected semicomplete c-partite digraph with c≥3 and letting v be a vertex of D,then Dhas a(c-1)-pan-outpath partly ordering from v.This result improves a theorem about outpaths in semicomplete multipartite digraphs obtained by Guo in 1999.     

Locally semicomplete digraphs: A generalization of tournaments

In this paper we introduce a new class of directed graphs called locally semicomplete digraphs. These are defined to be those digraphs for which the following holds: for every vertex x the vertices dominated by x induce a semicomplete digraph and the vertices that dominate x induce a semicomplete digraph. (A digraph is semicomplete if for any two distinct vertices u and ν, there is at least one arc between them.) This class contains the class of semicomplete digraphs, but is much more general. In fact, the class of underlying graphs of the locally semi-complete digraphs is precisely the class of proper circular-arc graphs (see [13], Theorem 3). We show that many of the classic theorems for tournaments have natural analogues for locally semicomplete digraphs. For example, every locally semicomplete digraph has a directed Hamiltonian path and every strong locally semicomplete digraph has a Hamiltonian cycle. We also consider connectivity properties, domination orientability, and algorithmic aspects of locally semicomplete digraphs. Some of the results on connectivity are new, even when restricted to semicomplete digraphs.     

On pancyclic digraphs

We show that a strongly connected digraph with n vertices and minimum degree ? n is pancyclic unless it is one of the graphs K_p,p. This generalizes a result of A. Ghouila-Houri. We disprove a conjecture of J. A. Bondy by showing that there exist hamiltonian digraphs with n vertices and $\text{1}\text{2}\text{n(n + 1) – 3}$ edges which are not pancyclic. We show that any hamiltonian digraph with n vertices and at least $\text{1}\text{2}\text{n(n + 1) – 1}$ edges is pancyclic and we give some generalizations of this result. As applications of these results we determine the minimal number of edges required in a digraph to guarantee the existence of a cycle of length k, k ? 2, and we consider the corresponding problem where the digraphs under consideration are assumed to be strongly connected.     

Connected -tuple twin domination in de Bruijn and Kautz digraphs

For a digraph G, a k-tuple twin dominating set D of G for some fixed k≥1 is a set of vertices such that every vertex is adjacent to at least k vertices in D, and also every vertex is adjacent from at least k vertices in D. If the subgraph of G induced by D is strongly connected, then D is called a connected k-tuple twin dominating set of G. In this paper, we give constructions of minimal connected k-tuple twin dominating sets for de Bruijn digraphs and Kautz digraphs.     

Double-super-connected digraphs

A strongly connected digraph D is said to be super-connected if every minimum vertex-cut is the out-neighbor or in-neighbor set of a vertex. A strongly connected digraph D is said to be double-super-connected if every minimum vertex-cut is both the out-neighbor set of a vertex and the in-neighbor set of a vertex. In this paper, we characterize the double-super-connected line digraphs, Cartesian product and lexicographic product of two digraphs. Furthermore, we study double-super-connected Abelian Cayley digraphs and illustrate that there exist double-super-connected digraphs for any given order and minimum degree.     

Finding cheapest cycles in vertex-weighted quasi-transitive and extended semicomplete digraphs

We consider the problem of finding a minimum cost cycle in a digraph with real-valued costs on the vertices. This problem generalizes the problem of finding a longest cycle and hence is NP-hard for general digraphs. We prove that the problem is solvable in polynomial time for extended semicomplete digraphs and for quasi-transitive digraphs, thereby generalizing a number of previous results on these classes. As a byproduct of our method we develop polynomial algorithms for the following problem: Given a quasi-transitive digraph D with real-valued vertex costs, find, for each j=1,2,…,|V(D)|, j disjoint paths P₁,P₂,…,P_j such that the total cost of these paths is minimum among all collections of j disjoint paths in D.     

Kernels by properly colored paths in arc-colored digraphs

A kernel by properly colored paths of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a properly colored directed path in $D$. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for digraphs with no intersecting cycles, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.     

Kings in semicomplete multipartite digraphs

A digraph obtained by replacing each edge of a complete p‐partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete p‐partite digraph, or just a semicomplete multipartite digraph. A semicomplete multipartite digraph with no cycle of length two is a multipartite tournament. In a digraph D, an r‐king is a vertex q such that every vertex in D can be reached from q by a path of length at most r. Strengthening a theorem by K. M. Koh and B. P. Tan (Discr Math 147 (1995), 171–183) on the number of 4‐kings in multipartite tournaments, we characterize semicomplete multipartite digraphs, which have exactly k 4‐kings for every k = 1, 2, 3, 4, 5. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 177‐183, 2000     

正圆有向图中的弧不相交的Hamilton路和圈

2012年,Bang-Jensen和Huang(J.Combin.Theory Ser.B.2012,102:701-714)证明了2-弧强的局部半完全有向图可以分解为两个弧不相交的强连通生成子图当且仅当D不是偶圈的二次幂,并提出了任意3-强的局部竞赛图中包含两个弧不相交的Hamilton圈的猜想.主要研究正圆有向图中的弧不相交的Hamilton路和Hamilton圈,并证明了任意3-弧强的正圆有向图中包含两个弧不相交的Hamilton圈和任意4-弧强的正圆有向图中包含一个Hamilton圈和两个Hamilton路,使得它们两两弧不相交.由于任意圆有向图一定是正圆有向图,所得结论可以推广到圆有向图中.又由于圆有向图是局部竞赛图的子图类,因此所得结论说明对局部竞赛图的子图类――圆有向图,Bang-Jensen和Huang的猜想成立.     

Directed distance in digraphs: Centers and medians

The directed distance d_D(u, v) from a vertex u to a vertex v in a strong digraph D is the length of a shortest (directed) u - v path in D. The eccentricity of a vertex v in D is the directed distance from v to a vertex furthest from v. The distance of a vertex v in D is the sum of the directed distances from v to the vertices of D. The center C(D) of D is the subdigraph induced by those vertices of minimum eccentricity, while the median M(D) of D is the subdigraph induced by those vertices of minimum distance. It is shown that for every two asymmetric digraphs D₁ and D₂, there exists a strong asymmetric digraph H such that C(H) ? D₁ and M(H) ? D₂, and where the directed distance from C(H) to M(H) and from M(H) to C(H) can be arbitrarily prescribed. Furthermore, if K is a nonempty asymmetric digraph isomorphic to an induced subdigraph of both D₁ and D₂, then there exists a strong asymmetric digraph F such that C(F) ? D₁, M(F) ? D₂ and C(F) ∩ M(F) ? K. © 1993 John Wiley & Sons, Inc.     

A Characterization of Locally Semicomplete CKI-digraphs

A digraph is locally-in semicomplete if for every vertex of D its in-neighborhood induces a semicomplete digraph and it is locally semicomplete if for every vertex of D the in-neighborhood and the out-neighborhood induces a semicomplete digraph. The locally semicomplete digraphs where characterized in 1997 by Bang-Jensen et al. and in 1998 Bang-Jensen and Gutin posed the problem if finding a kernel in a locally-in semicomplete digraph is polynomial or not. A kernel of a digraph is a set of vertices, which is independent and absorbent. A digraph D such that every proper induced subdigraph of D has a kernel is said to be critical kernel imperfect digraph (CKI-digraph) if the digraph D does not have a kernel. A digraph without an induced CKI-digraph as a subdigraph does have a kernel. We characterize the locally semicomplete digraphs, which are CKI. As a consequence of this characterization we conclude that determinate whether a locally semicomplete digraph is a CKI-digraph or not, is polynomial.     

On Vertices of outdegree n in minimally n‐connected digraphs

Let |D| and |D|⁺_n denote the number of vertices of D and the number of vertices of outdegree n in the digraph D, respectively. It is proved that every minimally n‐connected, finite digraph D has |D|⁺_n ≥ n + 1 and that for n ≥ 2, there is a c_n > 0 such that for all minimally n‐connected, finite digraphs D. Furthermore, case n = 2 of the following conjecture is settled which says that every minimally n‐connected, finite digraph has a vertex of indegree and outdegree equal to n. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 129–144, 2002     

Digraphs with the path-merging property

In this paper we study those digraphs D for which every pair of internally disjoint (X, Y)-paths P₁, P₂ can be merged into one (X, Y)-path P* such that V(P₁) ∪ V(P₂), for every choice of vertices X, Y ? V(D). We call this property the path-merging property and we call a graph path-mergeable if it has the path-merging property. We show that each such digraph has a directed hamiltonian cycle whenever it can possibly have one, i.e., it is strong and the underlying graph has no cutvertex. We show that path-mergeable digraphs can be recognized in polynomial time and we give examples of large classes of such digraphs which are not contained in any previously studied class of digraphs. We also discuss which undirected graphs have path-mergeable digraph orientations. © 1995, John Wiley & Sons, Inc.     

The Second Neighbourhood for Quasi-transitive Oriented Graphs

In 2006, Sullivan stated the conjectures:(1) every oriented graph has a vertex x such that d~(++)(x) ≥ d~-(x);(2) every oriented graph has a vertex x such that d~(++)(x) + d~+(x) ≥ 2 d~-(x);(3) every oriented graph has a vertex x such that d~(++)(x) + d~+(x) ≥ 2 · min{d~+(x), d~-(x)}. A vertex x in D satisfying Conjecture(i) is called a Sullivan-i vertex, i = 1, 2, 3. A digraph D is called quasi-transitive if for every pair xy, yz of arcs between distinct vertices x, y, z, xz or zx("or" is inclusive here) is in D. In this paper, we prove that the conjectures hold for quasi-transitive oriented graphs, which is a superclass of tournaments and transitive acyclic digraphs. Furthermore, we show that a quasi-transitive oriented graph with no vertex of in-degree zero has at least three Sullivan-1 vertices and a quasi-transitive oriented graph has at least three Sullivan-3 vertices unless it belongs to an exceptional class of quasitransitive oriented graphs. For Sullivan-2 vertices, we show that an extended tournament, a subclass of quasi-transitive oriented graphs and a superclass of tournaments, has at least two Sullivan-2 vertices unless it belongs to an exceptional class of extended tournaments.     

Sufficient Conditions for Super-Arc-Strongly Connected Oriented Graphs

A digraph D is called super-arc-strongly connected if the arcs of every its minimum arc-disconnected set are incident to or from some vertex in D. A digraph without any directed cycle of length 2 is called an oriented graph. Sufficient conditions for digraphs to be super-arc-strongly connected have been given by several authors. However, closely related conditions for super-arc-strongly connected oriented graphs have little attention until now. In this paper we present some minimum degree and degree sequence conditions for oriented graphs to be super-arc-strongly connected.     

Kernels by monochromatic paths in digraphs with covering number 2

We call the digraph D an k-colored digraph if the arcs of D are colored with k colors. A subdigraph H of D is called monochromatic if all of its arcs are colored alike. A set N⊆V(D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u,v∈N, there is no monochromatic directed path between them, and (ii) for every vertex x∈(V(D)?N), there is a vertex y∈N such that there is an xy-monochromatic directed path. In this paper, we prove that if D is an k-colored digraph that can be partitioned into two vertex-disjoint transitive tournaments such that every directed cycle of length 3,4 or 5 is monochromatic, then D has a kernel by monochromatic paths. This result gives a positive answer (for this family of digraphs) of the following question, which has motivated many results in monochromatic kernel theory: Is there a natural numberlsuch that if a digraphDisk-colored so that every directed cycle of length at mostlis monochromatic, thenDhas a kernel by monochromatic paths?     

Independent sets and non-augmentable paths in arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs

A digraph is arc-locally in-semicomplete if for any pair of adjacent vertices x,y, every in-neighbor of x and every in-neighbor of y either are adjacent or are the same vertex. A digraph is quasi-arc-transitive if for any arc xy, every in-neighbor of x and every out-neighbor of y either are adjacent or are the same vertex. Laborde, Payan and Xuong proposed the following conjecture: Every digraph has an independent set intersecting every non-augmentable path (in particular, every longest path). In this paper, we shall prove that this conjecture is true for arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs.     

On strong digraphs with a prescribed ultracenter

The (directed) distance from a vertex u to a vertex v in a strong digraph D is the length of a shortest u-v (directed) path in D. The eccentricity of a vertex v of D is the distance from v to a vertex furthest from v in D. The radius radD is the minimum eccentricity among the vertices of D and the diameter diamD is the maximum eccentricity. A central vertex is a vertex with eccentricity radD and the subdigraph induced by the central vertices is the center C(D). For a central vertex v in a strong digraph D with radD < diamD, the central distance c(v) of v is the greatest nonnegative integer n such that whenever d(v, x) n, then x is in C(D). The maximum central distance among the central vertices of D is the ultraradius uradD and the subdigraph induced by the central vertices with central distance uradD is the ultracenter UC(D). For a given digraph D, the problem of determining a strong digraph H with UC(H) = D and C(H) D is studied. This problem is also considered for digraphs that are asymmetric.