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The Dynamics of the Forest Graph Operator

Suresh Dara, S.M. Hegde, Venkateshwarlu Deva, S.B. Rao, Thomas Zaslavsky (2016)

Discussiones Mathematicae Graph Theory

In 1966, Cummins introduced the “tree graph”: the tree graph T(G) of a graph G (possibly infinite) has all its spanning trees as vertices, and distinct such trees correspond to adjacent vertices if they differ in just one edge, i.e., two spanning trees T1 and T2 are adjacent if T2 = T1 − e + f for some edges e ∈ T1 and f ∉ T1. The tree graph of a connected graph need not be connected. To obviate this difficulty we define the “forest graph”: let G be a labeled graph of order α, finite or infinite,...

The fundamental group of a locally finite graph with ends-a hyperfinite approach

Isaac Goldbring, Alessandro Sisto (2016)

Fundamenta Mathematicae

The end compactification |Γ| of a locally finite graph Γis the union of the graph and its ends, endowed with a suitable topology. We show that π₁(|Γ|) embeds into a nonstandard free group with hyperfinitely many generators, i.e. an ultraproduct of finitely generated free groups, and that the embedding we construct factors through an embedding into an inverse limit of free groups. We also show how to recover the standard description of π₁(|Γ|) given by Diestel and Sprüssel (2011). Finally, we give...

The Quest for A Characterization of Hom-Properties of Finite Character

Izak Broere, Moroli D.V. Matsoha, Johannes Heidema (2016)

Discussiones Mathematicae Graph Theory

A graph property is a set of (countable) graphs. A homomorphism from a graph G to a graph H is an edge-preserving map from the vertex set of G into the vertex set of H; if such a map exists, we write G → H. Given any graph H, the hom-property →H is the set of H-colourable graphs, i.e., the set of all graphs G satisfying G → H. A graph property P is of finite character if, whenever we have that F ∈ P for every finite induced subgraph F of a graph G, then we have that G ∈ P too. We explore some of...

Topological circles and Euler tours in locally finite graphs.

Georgakopoulos, Agelos (2009)

The Electronic Journal of Combinatorics [electronic only]

Travel groupoids on infinite graphs

Jung Rae Cho, Jeongmi Park, Yoshio Sano (2014)

Czechoslovak Mathematical Journal

The notion of travel groupoids was introduced by L. Nebeský in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set $V$ and a binary operation $*$ on $V$ satisfying two axioms. We can associate a graph with a travel groupoid. We say that a graph $G$ has a travel groupoid if the graph associated with the travel groupoid is equal to $G$ . Nebeský gave a characterization of finite graphs having a travel groupoid. In this paper, we study travel groupoids on infinite graphs....