On Vosperian and Superconnected Vertex-Transitive Digraphs

We investigate the structure of a digraph having a transitive automorphism group where every cutset of minimal cardinality consists of all successors or all predecessors of some vertex. We give a complete characterization of vosperian arc-transitive digraphs. It states that an arc-transitive strongly connected digraph is vosperian if and only if it is irreducible. In particular, this is the case if the degree is coprime with the order of the digraph. We give also a complete characterization of vosperian Cayley digraphs and a complete characterization of irreducible superconnected Cayley digraphs. These two last characterizations extend the corresponding ones in Abelian Cayley digraphs and the ones in the undirected case.     

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.     

Some constructions of highly arc-transitive digraphs

We construct infinite highly arc-transitive digraphs with finite out-valency and whose sets of descendants are digraphs which have a homomorphism onto a directed (rooted) tree. Some of these constructions are based on [4] and [5], and are shown to have universal reachability relation.     

On the structure of strong 3-quasi-transitive digraphs

In this paper, D=(V(D),A(D)) denotes a loopless directed graph (digraph) with at most one arc from u to v for every pair of vertices u and v of V(D). Given a digraph D, we say that D is 3-quasi-transitive if, whenever u→v→w→z in D, then u and z are adjacent or u=z. In Bang-Jensen (2004) [3], Bang-Jensen introduced 3-quasi-transitive digraphs and claimed that the only strong 3-quasi-transitive digraphs are the strong semicomplete digraphs and strong semicomplete bipartite digraphs. In this paper, we exhibit a family of strong 3-quasi-transitive digraphs distinct from strong semicomplete digraphs and strong semicomplete bipartite digraphs and provide a complete characterization of strong 3-quasi-transitive digraphs.     

关于交换群上的Cayley有向图的正规性

Cayley有向图X=Cay(G,S)叫做正规的,如果G的右正则表示R(G)在X的全自同构群Aut(X)中正规,我们定出了交换群上的小度数的非正规的Cayley有向图, 并给出了一个猜想．应用这个结果,给出了pn(n≤2)个点上的度数不超过3的有向对称图的分类,这里p是一个奇素数．     

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.     

Highly Arc-Transitive Digraphs With No Homomorphism Onto <Emphasis Type="Bold">Z</Emphasis>

In an infinite digraph D, an edge e' is reachable from an edge e if there exists an alternating walk in D whose initial and terminal edges are e and e'. Reachability is an equivalence relation and if D is 1-arc-transitive, then this relation is either universal or all of its equivalence classes induce isomorphic bipartite digraphs. In Combinatorica, 13 (1993), Cameron, Praeger and Wormald asked if there exist highly arc-transitive digraphs (apart from directed cycles) for which the reachability relation is not universal and which do not have a homomorphism onto the two-way infinite directed path (a Cayley digraph of Z with respect to one generator). In view of an earlier result of Praeger in Australas. J. Combin., 3 (1991), such digraphs are either locally infinite or have equal in- and out-degree. In European J. Combin., 18 (1997), Evans gave an affirmative answer by constructing a locally infinite example. For each odd integer n >= 3, a construction of a highly arc-transitive digraph without property Z satisfying the additional properties that its in- and out-degree are equal to 2 and that the reachability equivalence classes induce alternating cycles of length 2n, is given. Furthermore, using the line digraph operator, digraphs having the above properties but with alternating cycles of length 4 are obtained. Received April 12, 1999 Supported in part by "Ministrstvo za šolstvo, znanost in šport Slovenije", research program PO-0506-0101-99.     

Linear arboricity of regular digraphs

A linear directed forest is a directed graph in which every component is a directed path.The linear arboricity la(D) of a digraph D is the minimum number of linear directed forests in D whose union covers all arcs of D. For every d-regular digraph D, Nakayama and P′eroche conjecture that la(D) = d + 1. In this paper, we consider the linear arboricity for complete symmetric digraphs,regular digraphs with high directed girth and random regular digraphs and we improve some wellknown results. Moreover, we propose a more precise conjecture about the linear arboricity for regular digraphs.     

Graphs and digraphs with all 2-factors isomorphic

We show that a digraph which contains a directed 2-factor and has minimum in-degree and out-degree at least four has two non-isomorphic directed 2-factors. As a corollary, we deduce that every graph which contains a 2-factor and has minimum degree at least eight has two non-isomorphic 2-factors. In addition we construct: an infinite family of 3-diregular digraphs with the property that all their directed 2-factors are Hamilton cycles, an infinite family of 2-connected 4-regular graphs with the property that all their 2-factors are isomorphic, and an infinite family of cyclically 6-edge-connected cubic graphs with the property that all their 2-factors are Hamilton cycles.     

List coloring digraphs

The dichromatic number of a digraph D is the least number k such that the vertex set of D can be partitioned into k parts each of which induces an acyclic subdigraph. Introduced by Neumann‐Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this article, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number behave the same in many contexts, such as in small digraphs (by proving a directed version of Ohba's conjecture), tournaments, and random digraphs. We then consider bipartite digraphs, and show that their list dichromatic number can be as large as . We finally give a Brooks‐type upper bound on the list dichromatic number of digon‐free digraphs.     

Acyclic systems of representatives and acyclic colorings of digraphs

A natural digraph analog of the graph theoretic concept of “an independent set” is that of “an acyclic set of vertices,” namely a set not spanning a directed cycle. By this token, an analog of the notion of coloring of a graph is that of decomposition of a digraph into acyclic sets. We extend some known results on independent sets and colorings in graphs to acyclic sets and acyclic colorings of digraphs. In particular, we prove bounds on the topological connectivity of the complex of acyclic sets, and using them we prove sufficient conditions for the existence of acyclic systems of representatives of a system of sets of vertices. These bounds generalize a result of Tardos and Szabó. We prove a fractional version of a strong‐acyclic‐coloring conjecture for digraphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 177–189, 2008     

On the weak distance-regularity of Moore-type digraphs

We prove that Moore digraphs, and some other classes of extremal digraphs, are weakly distance-regular in the sense that there is an invariance of the number of walks between vertices at a given distance. As weakly distance-regular digraphs, we then compute their complete spectrum from a ‘small’ intersection matrix. This is a very useful tool for deriving some results about their existence and/or their structural properties. For instance, we present here an alternative and unified proof of the existence results on Moore digraphs, Moore bipartite digraphs and, more generally, Moore generalized p-cycles. In addition, we show that the line digraph structure appears as a characteristic property of any Moore generalized p-cycle of diameter D?≥?2p.     

On extremal weighted digraphs with no heavy paths

Bollobás and Scott proved that if the weighted outdegree of every vertex of an edge-weighted digraph is at least 1, then the digraph contains a (directed) path of weight at least 1. In this note we characterize the extremal weighted digraphs with no heavy paths. Our result extends a corresponding theorem of Bondy and Fan on weighted graphs. We also give examples to show that a result of Bondy and Fan on the existence of heavy paths connecting two given vertices in a 2-connected weighted graph does not extend to 2-connected weighted digraphs.     

Cliques that are tolerance digraphs

In this paper, we study directed graph versions of tolerance graphs, in particular, the class of totally bounded bitolerance digraphs and several subclasses. When the underlying graph is complete, we prove that the classes of totally bounded bitolerance digraphs and interval catch digraphs are equal, and this implies a polynomial-time recognition algorithm for the former class. In addition, we give examples (whose underlying graphs are complete) to separate every other pair of subclasses, and one of these provides a counterexample to a conjecture of Maehara (1984).     

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.     

Finding a subdivision of a prescribed digraph of order 4

The problem of when a given digraph contains a subdivision of a fixed digraph F is considered. Bang‐Jensen et al. [4] laid out foundations for approaching this problem from the algorithmic point of view. In this article, we give further support to several open conjectures and speculations about algorithmic complexity of finding F‐subdivisions. In particular, up to five exceptions, we completely classify for which 4‐vertex digraphs F, the F‐subdivision problem is polynomial‐time solvable and for which it is NP‐complete. While all NP‐hardness proofs are made by reduction from some version of the 2‐linkage problem in digraphs, some of the polynomial‐time solvable cases involve relatively complicated algorithms.     

有向P2-路图的结构性质

一个有向图D的有向Pk-路图Pk(D)是通过把D中的所有有向k长路作为点集;两点u= x1x2…xk+1,v=y1y2…yk+1之间有弧uv当xi=yi-1,i=2,3,…,k+1.明显地,当k=1时Pk(D)就是通常的有向线图L(D).在[1,2]中,P2-路图得到完整刻画.在[3]中,Broersma等人研究了有向...     

Kings in locally semicomplete digraphs

A k‐king in a digraph D is a vertex which can reach every other vertex by a directed path of length at most k. We consider k‐kings in locally semicomplete digraphs and mainly prove that all strong locally semicomplete digraphs which are not round decomposable contain a 2‐king. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 279–287, 2010     

On finite reflexive homomorphism-homogeneous binary relational systems

A structure is called homogeneous if every isomorphism between finitely induced substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Nešet?il introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely induced substructures of the structure extends to an endomorphism of the structure.In this paper, we consider finite homomorphism-homogeneous relational systems with one reflexive binary relation. We show that for a large part of such relational systems (bidirectionally connected digraphs; a digraph is bidirectionally connected if each of its connected components can be traversed by ?-paths) the problem of deciding whether the system is homomorphism-homogeneous is coNP-complete. Consequently, for this class of relational systems there is no polynomially computable characterization (unless P=NP). On the other hand, in case of bidirectionally disconnected digraphs we present the full characterization. Our main result states that if a digraph is bidirectionally disconnected, then it is homomorphism-homogeneous if and only if it is either a finite homomorphism-homogeneous quasiorder, or an inflation of a homomorphism-homogeneous digraph with involution (a specific class of digraphs introduced later in the paper), or an inflation of a digraph whose only connected components are and .