Locally-finite connected-homogeneous digraphs

A digraph is connected-homogeneous if any isomorphism between finite connected induced subdigraphs extends to an automorphism of the digraph. We consider locally-finite connected-homogeneous digraphs with more than one end. In the case that the digraph embeds a triangle we give a complete classification, obtaining a family of tree-like graphs constructed by gluing together directed triangles. In the triangle-free case we show that these digraphs are highly arc-transitive. We give a classification in the two-ended case, showing that all examples arise from a simple construction given by gluing along a directed line copies of some fixed finite directed complete bipartite graph. When the digraph has infinitely many ends we show that the descendants of a vertex form a tree, and the reachability graph (which is one of the basic building blocks of the digraph) is one of: an even cycle, a complete bipartite graph, the complement of a perfect matching, or an infinite semiregular tree. We give examples showing that each of these possibilities is realised as the reachability graph of some connected-homogeneous digraph, and in the process we obtain a new family of highly arc-transitive digraphs without property Z.     

Arc transitive covering digraphs and their eigenvalues

We prove that every finite regular digraph has an arc-transitive covering digraph (whose arcs are equivalent under automorphisms) and every finite regular graph has a 2-arc-transitive covering graph. As a corollary, we sharpen C. D. Godsil's results on eigenvalues and minimum polynomials of vertex-transitive graphs and digraphs. Using Godsil's results, we prove, that given an integral matrix A there exists an arc-transitive digraph X such that the minimum polynomial of A divides that of X. It follows that there exist arc-transitive digraphs with nondiagonalizable adjacency matrices, answering a problem by P. J. Cameron. For symmetric matrices A, we construct a 2-arc-transitive graphs X.     

Descendant-homogeneous digraphs

The descendant setdesc(α) of a vertex α in a digraph D is the set of vertices which can be reached by a directed path from α. A subdigraph of D is finitely generated if it is the union of finitely many descendant sets, and D is descendant-homogeneous if it is vertex transitive and any isomorphism between finitely generated subdigraphs extends to an automorphism. We consider connected descendant-homogeneous digraphs with finite out-valency, specially those which are also highly arc-transitive. We show that these digraphs must be imprimitive. In particular, we study those which can be mapped homomorphically onto Z and show that their descendant sets have only one end.There are examples of descendant-homogeneous digraphs whose descendant sets are rooted trees. We show that these are highly arc-transitive and do not admit a homomorphism onto Z. The first example (Evans (1997) [6]) known to the authors of a descendant-homogeneous digraph (which led us to formulate the definition) is of this type. We construct infinitely many other descendant-homogeneous digraphs, and also uncountably many digraphs whose descendant sets are rooted trees but which are descendant-homogeneous only in a weaker sense, and give a number of other examples.     

Some sufficient conditions for the existence of kernels in infinite digraphs

A kernel N of a digraph D is an independent set of vertices of D such that for every w∈V(D)−N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. D is called a critical kernel imperfect digraph when D has no kernel but every proper induced subdigraph of D has a kernel. If F is a set of arcs of D, a semikernel modulo F of D is an independent set of vertices S of D such that for every z∈V(D)−S for which there exists an (S,z)-arc of D−F, there also exists an (z,S)-arc in D. In this work we show sufficient conditions for an infinite digraph to be a kernel perfect digraph, in terms of semikernel modulo F. As a consequence it is proved that symmetric infinite digraphs and bipartite infinite digraphs are kernel perfect digraphs. Also we give sufficient conditions for the following classes of infinite digraphs to be kernel perfect digraphs: transitive digraphs, quasi-transitive digraphs, right (or left)-pretransitive digraphs, the union of two right (or left)-pretransitive digraphs, the union of a right-pretransitive digraph with a left-pretransitive digraph, the union of two transitive digraphs, locally semicomplete digraphs and outward locally finite digraphs.     

Infinite highly arc transitive digraphs and universal covering digraphs

A digraph (that is a directed graph) is said to be highly arc transitive if its automorphism group is transitive on the set ofs-arcs for eachs0. Several new constructions are given of infinite highly arc transitive digraphs. In particular, for a connected, 1-arc transitive, bipartite digraph, a highly arc transitive digraphDL() is constructed and is shown to be a covering digraph for every digraph in a certain classD() of connected digraphs. Moreover, if is locally finite, thenDL() is a universal covering digraph forD(). Further constructions of infinite highly arc transitive digraphs are given.The second author wishes to acknowledge the hospitality of the Mathematical Institute of the University of Oxford, and the University of Auckland, during the period when the research for this paper was doneResearch supported by the Australian Research Council     

Exponents of a class of two-colored digraphs

A two-colored digraph D is primitive if there exist nonnegative integers h and k with h+k>0 such that for each pair (i, j) of vertices there exists an (h, k)-walk in D from i to j. The exponent of the primitive two-colored digraph D is the minimum value of h+k taken over all such h and k. In this article, we consider special primitive two-colored digraphs whose uncolored digraph has n+s vertices and consist of one n-cycle and one (n???2)-cycle. We give the bounds on the exponents, and the characterizations of the extremal two-colored digraphs.     

Quasi-transitive digraphs

A digraph is quasi-transitive if there is a complete adjacency between the inset and the outset of each vertex. Quasi-transitive digraphs are interseting because of their relation to comparability graphs. Specifically, a graph can be oriented as a quasi-transitive digraph if and only if it is a comparability graph. Quasi-transitive digraphs are also of interest as they share many nice properties of tournaments. Indeed, we show that every strongly connected quasi-transitive digraphs D on at least four vertices has two vertices v₁ and v₂ such that D – v_i is strongly connected for i = 1, 2. A result of tournaments on the existence of a pair of arc-disjoint in- and out-branchings rooted at the same vertex can also be extended to quasi-transitive digraphs. However, some properties of tournaments, like hamiltonicity, cannot be extended directly to quasi-transitive digraphs. Therefore we characterize those quasi-transitive digraphs which have a hamiltonian cycle, respectively a hamiltonian path. We show the existence of highly connected quasi-transitive digraphs D with a factor (a collection of disjoint cycles covering the vertex set of D), which have a cycle of every length 3 ≦ k ≦ |V(D)| ? 1 through every vertex and yet they are not hamiltonian. Finally we characterize pancyclic and vertex pancyclic quasi-transitive digraphs. © 1995, John Wiley & Sons, Inc.     

A lower bound for the spectral radius of a digraph

We show that the spectral radius ρ(D) of a digraph D with n vertices and c₂ closed walks of length 2 satisfies Moreover, equality occurs if and only if D is the symmetric digraph associated to a -regular graph, plus some arcs that do not belong to cycles. As an application of this result, we construct new sharp upper bounds for the low energy of a digraph, which extends Koolen and Moulton bounds of the energy to digraphs.     

The skew energy of a digraph

We are interested in the energy of the skew-adjacency matrix of a directed graph D, which is simply called the skew energy of D in this paper. Properties of the skew energy of D are studied. In particular, a sharp upper bound for the skew energy of D is derived in terms of the order of D and the maximum degree of its underlying undirected graph. An infinite family of digraphs attaining the maximum skew energy is constructed. Moreover, the skew energy of a directed tree is independent of its orientation, and interestingly it is equal to the energy of the underlying undirected tree. Skew energies of directed cycles under different orientations are also computed. Some open problems are presented.     

Perfect Digraphs

The clique number of a digraph D is the size of the largest bidirectionally complete subdigraph of D. D is perfect if, for any induced subdigraph H of D, the dichromatic number defined by Neumann‐Lara (The dichromatic number of a digraph, J. Combin. Theory Ser. B 33 (1982), 265–270) equals the clique number . Using the Strong Perfect Graph Theorem (M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann. Math. 164 (2006), 51–229) we give a characterization of perfect digraphs by a set of forbidden induced subdigraphs. Modifying a recent proof of Bang‐Jensen et al. (Finding an induced subdivision of a digraph, Theoret. Comput. Sci. 443 (2012), 10–24) we show that the recognition of perfect digraphs is co‐‐complete. It turns out that perfect digraphs are exactly the complements of clique‐acyclic superorientations of perfect graphs. Thus, we obtain as a corollary that complements of perfect digraphs have a kernel, using a result of Boros and Gurvich (Perfect graphs are kernel solvable, Discrete Math. 159 (1996), 35–55). Finally, we prove that it is ‐complete to decide whether a perfect digraph has a kernel.     

The gallai-younger conjecture for planar graphs

Younger conjectured that for everyk there is ag(k) such that any digraphG withoutk vertex disjoint cycles contains a setX of at mostg(k) vertices such thatG–X has no directed cycles. Gallai had previously conjectured this result fork=1. We prove this conjecture for planar digraphs. Specifically, we show that ifG is a planar digraph withoutk vertex disjoint directed cycles, thenG contains a set of at mostO(klog(k)log(log(k))) vertices whose removal leaves an acyclic digraph. The work also suggests a conjecture concerning an extension of Vizing's Theorem for planar graphs.     

Spectral radius of digraphs with given dichromatic number

Let D be a digraph with vertex set V(D). A partition of V(D) into k acyclic sets is called a k-coloring of D. The minimum integer k for which there exists a k-coloring of D is the dichromatic number χ(D) of the digraph D. Denote G_n,k the set of the digraphs of order n with the dichromatic number k≥2. In this note, we characterize the digraph which has the maximal spectral radius in G_n,k. Our result generalizes the result of [8] by Feng et al.     

Coefficients of ergodicity and the scrambling index

For a primitive stochastic matrix S, upper bounds on the second largest modulus of an eigenvalue of S are very important, because they determine the asymptotic rate of convergence of the sequence of powers of the corresponding matrix. In this paper, we introduce the definition of the scrambling index for a primitive digraph. The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). We investigate the scrambling index for primitive digraphs, and give an upper bound on the scrambling index of a primitive digraph in terms of the order and the girth of the digraph. By doing so we provide an attainable upper bound on the second largest modulus of eigenvalues of a primitive matrix that make use of the scrambling index.     

Primitive digraphs with the largest scrambling index

The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). In [M. Akelbek, S. Kirkland, Coefficients of ergodicity and the scrambling index, preprint] we gave the upper bound on k(D) in terms of the order and the girth of a primitive digraph D. In this paper, we characterize all the primitive digraphs such that the scrambling index is equal to the upper bound.     

Multiplicativity of acyclic local tournaments

A homomorphism of a digraph to another digraph is an edge-preserving vertex mapping. A digraphH is said to be multiplicative if the set of digraphs which do not admit a homomorphism toH is closed under categorical product. In this paper we discuss the multiplicativity of acyclic Hamiltonian digraphs, i.e., acyclic digraphs which contains a Hamiltonian path. As a consequence, we give a complete characterization of acyclic local tournaments with respect to multiplicativity.     

Decomposing <Emphasis Type="Italic">k</Emphasis>-ARc-Strong Tournaments Into Strong Spanning Subdigraphs

  The so-called Kelly conjecture states that every regular tournament on 2k+1 vertices has a decomposition into k-arc-disjoint hamiltonian cycles. In this paper we formulate a generalization of that conjecture, namely we conjecture that every k-arc-strong tournament contains k arc-disjoint spanning strong subdigraphs. We prove several results which support the conjecture:If D = (V, A) is a 2-arc-strong semicomplete digraph then it contains 2 arc-disjoint spanning strong subdigraphs except for one digraph on 4 vertices.Every tournament which has a non-trivial cut (both sides containing at least 2 vertices) with precisely k arcs in one direction contains k arc-disjoint spanning strong subdigraphs. In fact this result holds even for semicomplete digraphs with one exception on 4 vertices.Every k-arc-strong tournament with minimum in- and out-degree at least 37k contains k arc-disjoint spanning subdigraphs H ₁, H ₂, . . . , H _k such that each H _i is strongly connected.The last result implies that if T is a 74k-arc-strong tournament with speci.ed not necessarily distinct vertices u ₁, u ₂, . . . , u _k , v ₁, v ₂, . . . , v _k then T contains 2k arc-disjoint branchings where is an in-branching rooted at the vertex u _i and is an out-branching rooted at the vertex v _i , i=1,2, . . . , k. This solves a conjecture of Bang-Jensen and Gutin [3].We also discuss related problems and conjectures.

On the sum of the total domination numbers of a digraph and its converse

A vertex subset S of a digraph D is called a dominating set of D if every vertex not in S has an in-neighbor in S. A dominating set S of D is called a total dominating set of D if the subdigraph induced by S has no isolated vertices. The total domination number of D, denoted by γt(D), is the minimum cardinality of a total dominating set of D. We show that for any connected digraph D of order n≥3, γt(D)+γt(D^? )≤5n/3, where D? is the converse of D. Furthermore, we characterize the oriented trees for which the equality holds.     

(p,q)-odd digraphs

A digraph D is (p,q)-odd if and only if any subdivision of D contains a directed cycle of length different from p mod q. A characterization of (p,q)-odd digraphs analogous to the Seymour-Thomassen characterization of (1, 2)-odd digraphs is provided. In order to obtain this characterization we study the lattice generated by the directed cycles of a strongly connected digraph. We show that the sets of directed cycles obtained from an ear decomposition of the digraph in a natural way are bases of this lattice. A similar result does not hold for undirected graphs. However we construct, for each undirected 2-connected graph G, a set of cycles of G which form a basis of the lattice generated by the cycles of G. © 1996 John Wiley & Sons, Inc.     

The elimination procedure for the competition number is not optimal

Given an acyclic digraph D, the competition graph C(D) is defined to be the undirected graph with V(D) as its vertex set and where vertices x and y are adjacent if there exists another vertex z such that the arcs (x,z) and (y,z) are both present in D. The competition number k(G) for an undirected graph G is the least number r such that there exists an acyclic digraph F on |V(G)|+r vertices where C(F) is G along with r isolated vertices. Kim and Roberts [The Elimination Procedure for the Competition Number, Ars Combin. 50 (1998) 97-113] introduced an elimination procedure for the competition number, and asked whether the procedure calculated the competition number for all graphs. We answer this question in the negative by demonstrating a graph where the elimination procedure does not calculate the competition number. This graph also provides a negative answer to a similar question about the related elimination procedure for the phylogeny number introduced by the current author in [S.G. Hartke, The Elimination Procedure for the Phylogeny Number, Ars Combin. 75 (2005) 297-311].     

Generalized competition index of a primitive digraph

For positive integers k and m, and a digraph D, the k-step m-competition graph of D has the same set of vertices as D and an edge between vertices x and y if and only if there are distinct m vertices v₁,v₂,…,v_m in D such that there are directed walks of length k from x to v_i and from y to v_i for 1?i?m. In this paper, we present the definition of m-competition index for a primitive digraph. The m-competition index of a primitive digraph D is the smallest positive integer k such that is a complete graph. We study m-competition indices of primitive digraphs and provide an upper bound for the m-competition index of a primitive digraph.