Infinite directed paths in locally finite digraphs

Locally-finite connected-homogeneous digraphs

On locally finite transitive two - ended digraphs *

Since decades transitive graphs are a topic of great interest. The study of s-edge transitive (undirected) graphs goes back to Tutte [13], who showed that finite cubic graphs cannot be s-edge transitive for s> 5. Weiss [14] proved several years later that the only finite connected s-edge transitive graphs with s = 8 are the cycles. Considering directed graphs the situation is much more involved...

Edmonds' Branching Theorem in Digraphs Without Forward-Infinite Paths

Let D be a finite digraph, and let V0, . . . , Vk−1 be nonempty subsets of V (D). The (strong form of) Edmonds’ branching theorem states that there are pairwise edge-disjoint spanning branchings B0, . . . ,Bk−1 in D such that the root set of Bi is Vi (i = 0, . . . , k − 1) if and only if for all ∅ 6= X ⊆ V (D) the number of ingoing edges of X is greater than or equal to the number of sets Vi di...

Distance Sequences In Locally Infinite Vertex-Transitive Digraphs

We prove that the out-distance sequence {f(k)} of a vertex-transitive digraph of finite or infinite degree satisfies f(k + 1) ≤ f(k) for k ≥ 1, where f(k) denotes the number of vertices at directed distance k from a given vertex. As a corollary, we prove that for a connected vertextransitive undirected graph of infinite degree d, we have f(k) = d for all k, 1 ≤ k < diam(G). This answers a quest...

Infinite Paths in Locally Finite Graphs and in Their Spanning Trees

The paper concerns infinite paths (in particular, the maximum number of pairwise vertex-disjoint ones) in locally finite graphs and in spanning trees of such graphs.