Countable Digraph Research Articles

Majority choosability of countable graphs

In any vertex coloring of a graph some edges have differently colored ends (bichromatic edges) and some are monochromatic. In a proper coloring all edges are bichromatic. In a majority coloring it is enough that for every vertex v, the number of monochromatic edges incident to v does not exceed the number of bichromatic edges incident to v. A well known result proved by Lovász asserts that every finite graph has a majority 2-coloring. A similar statement for countably infinite graphs is a challenging open problem, known as the Unfriendly Partition Conjecture.We consider a natural list variant of majority coloring. A graph is majorityk-choosable if it has a majority coloring from any lists of size k assigned arbitrarily to the vertices. We prove that every countable graph is majority 4-choosable. We also consider a natural analog of majority coloring for directed graphs. We prove that every countable digraph is also majority 4-choosable. We pose list and directed analogs of the Unfriendly Partition Conjecture, stating that every countable graph is majority 2-choosable and every countable digraph is majority 3-choosable.

Three conjectures of Ostrander on digraph Laplacian eigenvectors

Ostrander proposed three conjectures on the connections between topological properties of a weighted digraph and combinatorial properties of its Laplacian eigenvectors. We verify one of his conjectures, give counterexamples to the other two, and then seek for related valid connections and generalizations to Schrodinger operators on countable digraphs. We suggest the open question of deciding if the countability assumption can be dropped from our main results.

Countable Menger's Theorem with Finitary Matroid Constraints on the Ingoing Edges

We present a strengthening of the countable Menger's theorem of R. Aharoni. Let $ D=(V,A) $ be a countable digraph with $ s\neq t\in V $ and let $\mathcal{M}=\bigoplus_{v\in V}\mathcal{M}_v $ be a matroid on $ A $ where $ \mathcal{M}_v $ is a finitary matroid on the ingoing edges of $ v $. We show that there is a system of edge-disjoint $ s \rightarrow t $ paths $ \mathcal{P} $ such that the united edge set of these paths is $ \mathcal{M} $-independent, and there is a $ C \subseteq A $ consisting of one edge from each element of $\mathcal{P} $ for which $ \mathsf{span}_{\mathcal{M}}(C) $ covers all the $ s\rightarrow t $ paths in $ D $.

Infinite primitive and distance transitive directed graphs of finite out-valency

We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph.