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Algebraic approach to locally finite trees with one end

Bohdan Zelinka (2003)

Mathematica Bohemica

Let T be an infinite locally finite tree. We say that T has exactly one end, if in T any two one-way infinite paths have a common rest (infinite subpath). The paper describes the structure of such trees and tries to formalize it by algebraic means, namely by means of acyclic monounary algebras or tree semilattices. In these algebraic structures the homomorpisms and direct products are considered and investigated with the aim of showing, whether they give algebras with the required properties. At...

Analogues of cliques for oriented coloring

William F. Klostermeyer, Gary MacGillivray (2004)

Discussiones Mathematicae Graph Theory

We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.

Antisymmetric flows and strong colourings of oriented graphs

J. Nešetřill, André Raspaud (1999)

Annales de l'institut Fourier

The homomorphisms of oriented or undirected graphs, the oriented chromatic number, the relationship between acyclic colouring number and oriented chromatic number, have been recently intensely studied. For the purpose of duality, we define the notions of strong-oriented colouring and antisymmetric-flow. An antisymmetric-flow is a flow with values in an additive abelian group which uses no opposite elements of the group. We prove that the strong-oriented chromatic number χ → s (as the modular version...

Band combinatorics of domestic string algebras

Gena Puninski (2007)

Colloquium Mathematicae

We prove that the multiplicity of a simple module as a composition factor in a composition series for a primitive band module over a domestic string algebra is at most two.

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