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     

广义de Bruijn有向图及其叠线图的支撑树与欧拉环游的计数

给出了一类特殊的广义deBruijn有向图的支撑树与欧环游的数目的简洁表示式，并得到了广义deBruijn有向叠线图的支撑树与欧拉环境数目的计算公式。     

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.     

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.     

Hamiltonicity of large generalized de Bruijn cycles

In this article, we determine when the large generalized de Bruijn cycles are Hamiltonian. These digraphs have been introduced by Gómez, Padró and Pérennes as large interconnection networks with small diameter and they are a family of generalized cycles. They are Kronecker products of generalized de Bruijn digraphs and dicycles.     

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.     

A note on total colourings of digraphs

In the paper two different arc-colourings and two associated with the total colourings of digraphs are considered. In one of these colourings we show that the problem of calculating the total chromatic index reduces to that of calculating the chromatic number of the underlying graph. In the other colouring we find the total chromatic indices of complete symmetric digraphs and tournaments.     

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.     

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.     

Quadratic assignment problems on series-parallel digraphs

Christofides has shown that the quadratic assignment problem (QAP) on isomorphic trees is solvable in polynomial time by dynamic programming. We generalize the approach to minimal series-parallel digraphs and show that this problem is equivalent to the general QAP, thus NP-hard. However the restriction to the subclass of series-parallel digraphs not containing bipartite subgraphs again is solvable in polynomial time. An algorithm for this class is given.This work was financially supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, Projekt S32/01.