Infinite Digraph Research Articles

On Partitioning the Edges of an Infinite Digraph into Directed Cycles

Nash-Williams proved in [1] that for an undirected graph G the set E¹Gº can be partitioned into cycles if and only if there is no finite cut of odd size. Later C. Thomassen gave a simpler proof for this in [2] and conjectured the following directed analogue of the theorem: the edge set of a digraph can be partitioned into directed cycles if and only if for each subset of the vertices the cardinality of the ingoing and the outgoing edges are equal. The aim of the paper is to prove this conjecture.

Generating Infinite Digraphs by Derangements

Abstract A set $\mathcal{S}$ of derangements (fixed-point-free permutations) of a set V generates a digraph with vertex set V and arcs $(x,x^{\,\sigma})$ for x ∈ V and $\sigma\in\mathcal{S}$. We address the problem of characterizing those infinite (simple loopless) digraphs which are generated by finite sets of derangements. The case of finite digraphs was addressed in an earlier work by the second and third authors. A criterion is given for derangement generation which resembles the criterion given by De Bruijn and Erdős for vertex colourings of graphs in that the property for an infinite digraph is determined by properties of its finite sub-digraphs. The derangement generation property for a digraph is linked with the existence of a finite 1-factor cover for an associated bipartite (undirected) graph.

Connectedness of digraphs from quadratic polynomials

Suppose that f(x)=x(x−k), where k is an odd positive integer. First, an infinite digraph Gk=(V,E) is defined, where the vertex set is V=ℤ and the edge set is E={(x,y)∣x,y∈ℤ,f(x)=f(2y)}. Then the following results are proved: if k=1, then the digraph Gk is weakly connected; if p is a safe prime, i.e., both p and q=(p−1)∕2 are primes, then the number wp of weakly connected components of the digraph Gp is 2. Finally, a conjecture that there are infinitely many primes p such that wp=2 is presented.

Infinite kernel perfect digraphs

Let D be a digraph, possibly infinite, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. A subset K of V(D) is said to be a kernel if it is both independent (a vertex in K has no successor in K) and absorbing (a vertex not in K has a successor in K). An infinite digraph D is said to be a finitely critical kernel imperfect digraph if D contains no kernel but every finite induced subdigraph of D contains a kernel. In this paper we will characterize the infinite kernel perfect digraphs by means of finitely critical imperfect digraphs and strong components of its asymmetric part and then, by using some previous theorems for infinite digraphs, we will deduce several results from the main result. Richardson’s theorem establishes that if D is a finite digraph without cycles of odd length, then D has a kernel. In this paper we will show a generalization of this theorem for infinite digraphs.

Independent and monochromatic absorbent sets in infinite digraphs

Let D be a digraph, we say that it is an m-coloured digraph if the arcs of D are coloured with at most m-colours. An (u,v) arc is symmetrical if (v,u) is also an arc of D. A directed path (resp. directed cycle) is monochromatic if all of its arcs are coloured with the same colour, and it is quasi-monochromatic if at most one of its arcs is coloured with different colour.A set AN⊂V(D) is an A-kernel if it satisfies the following conditions: (1)For every pair of vertices x,y∈AN, there is no arc between them, we say that AN is independent.(2)For every x∉AN, there exists an xy-monochromatic path for some y∈AN, it means AN is absorbent by monochromatic paths. An infinite sequence of different vertices (x1,x2,x3,…) such that (xi,xi+1)∈A(D) for every i∈N will be called an infinite outward path.In this paper we introduce the definitions of A-kernel and A-semikernel, but also we prove the following theorem: Let D be a possibly infinite digraph, if every directed cycle and every infinite outward path has two consecutive vertices, say xi and xi+1, such that there exists an xi+1xi-monochromatic path, then D has an A-kernel.

Restricted domination in arc-colored digraphs

Let H = (V(H), A(H)) be a digraph possibly with loops and D = (V(D), A(D)) a digraph whose arcs are colored with the vertices of H (this is what we call an H-colored digraph); i.e. there exists a function c: A(D) → V(H); for an arc of D, f = (u, v) ∊ A(D), we call c(f) = c(u, v) the color of f. A directed walk (directed path) P = (u0, u1,…, un) in D will be called an H-walk (H-path) whenever (c(u0, u1), c(u1, u2),…, c(un-2, un-1), c(un-1, un)) is a directed walk (directed path) in H. We introduce the concept of H-kernel N, as a generalization of the two properties that define a kernel (Recall that a kernel N of a digraph D is a set of vertices N ⊆ V (D) which is independent and for each x ∊ V(D) — N, there exists an xN-arc in D). A set N ⊆ V(D) is called H-independent whenever for every two different vertices x, y ∊ N there is no H-path between them, and N is called H-absorbent whenever for each x ∊ V (D) — N there exists a vertex y ∊ N and an xy-H-path in D. The set N ⊆ V(D) will be called H-kernel if and only if it is H-independent and H-absorbent.This new concept generalizes the concepts of kernel, kernel by monochromatic paths and kernel by alternating paths.In this paper we show sufficient conditions for an infinite digraph to have an H-kernel.

Menger’s theorem for infinite graphs

We prove that Menger’s theorem is valid for infinite graphs, in the following strong version: let A and B be two sets of vertices in a possibly infinite digraph. Then there exist a set \(\mathcal{P}\) of disjoint A–B paths, and a set S of vertices separating A from B, such that S consists of a choice of precisely one vertex from each path in \(\mathcal{P}\). This settles an old conjecture of Erdős.

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.

Highly Arc-Transitive Digraphs With No Homomorphism Onto Z

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.

Compactness and finite equivalence of infinite digraphs

In Bauslaugh (1995) we defined and explored the notion of homomorphic compactness for infinite digraphs. In this paper we show that there are exactly 2 N 0 compact digraphs, up to homomorphic equivalence. We then define the notion of finite equivalence for infinite digraphs. We show that for almost any infinite digraph G, the class of digraphs which are finitely equivalent to G (modulo homomorphic equivalence) is either a proper class or consists of a single homomorphic equivalence class. For undirected graphs we show that this is true in all cases. We also examine some basic properties of a natural partial order we may impose on these classes of digraphs.

Infinite digraphs with nonreconstructible outvalency sequences

AbstractWe show that the outvalency sequence of an infinite digraph is not, in general, reconstructible.

A characterization of interval catch digraphs

It is shown that a (finite or infinite) digraph D is the catch digraph of a family of pointed intervals if and only if it contains no set of three vertices which fulfil: any two of them are weakly connected by a chain where no initial endpoint of an arc precedes the third vertex. Furthermore, it is shown that all these digraphs have weakly triangulated (and thus perfect) underlying graphs.

A context-free language and enumeration problems on infinite trees and digraphs

There are given: an infinite tree T r , an infinite digraph D r , and a context-free grammar G r . Then the relations between four problems are investigated: in T r , counting subtrees containing k edges (one of them fixed); in D r , counting paths of length 2 rk + 1; in the language generated by G r , counting the words of length 2 rk + 1; and in T 2, counting the subtrees satisfying a certain condition.